Properties

Label 2760.1.co.d.1259.1
Level $2760$
Weight $1$
Character 2760.1259
Analytic conductor $1.377$
Analytic rank $0$
Dimension $10$
Projective image $D_{22}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,1,Mod(419,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 11, 11, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.419");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2760.co (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: \(1.37741943487\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 1259.1
Root \(0.142315 - 0.989821i\) of defining polynomial
Character \(\chi\) \(=\) 2760.1259
Dual form 2760.1.co.d.1859.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.654861 - 0.755750i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-0.142315 - 0.989821i) q^{4} +(-0.841254 + 0.540641i) q^{5} +(-0.841254 + 0.540641i) q^{6} +(-0.841254 - 0.540641i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\) \(q+(0.654861 - 0.755750i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-0.142315 - 0.989821i) q^{4} +(-0.841254 + 0.540641i) q^{5} +(-0.841254 + 0.540641i) q^{6} +(-0.841254 - 0.540641i) q^{8} +(0.841254 + 0.540641i) q^{9} +(-0.142315 + 0.989821i) q^{10} +(-0.142315 + 0.989821i) q^{12} +(0.959493 - 0.281733i) q^{15} +(-0.959493 + 0.281733i) q^{16} +(-1.14231 - 0.989821i) q^{17} +(0.959493 - 0.281733i) q^{18} +(-1.49611 + 1.29639i) q^{19} +(0.654861 + 0.755750i) q^{20} +(0.415415 + 0.909632i) q^{23} +(0.654861 + 0.755750i) q^{24} +(0.415415 - 0.909632i) q^{25} +(-0.654861 - 0.755750i) q^{27} +(0.415415 - 0.909632i) q^{30} +(0.557730 + 1.89945i) q^{31} +(-0.415415 + 0.909632i) q^{32} +(-1.49611 + 0.215109i) q^{34} +(0.415415 - 0.909632i) q^{36} +1.97964i q^{38} +1.00000 q^{40} -1.00000 q^{45} +(0.959493 + 0.281733i) q^{46} +1.81926i q^{47} +1.00000 q^{48} +(-0.959493 - 0.281733i) q^{49} +(-0.415415 - 0.909632i) q^{50} +(0.817178 + 1.27155i) q^{51} +(-0.118239 + 0.822373i) q^{53} -1.00000 q^{54} +(1.80075 - 0.822373i) q^{57} +(-0.415415 - 0.909632i) q^{60} +(0.797176 - 0.234072i) q^{61} +(1.80075 + 0.822373i) q^{62} +(0.415415 + 0.909632i) q^{64} +(-0.817178 + 1.27155i) q^{68} +(-0.142315 - 0.989821i) q^{69} +(-0.415415 - 0.909632i) q^{72} +(-0.654861 + 0.755750i) q^{75} +(1.49611 + 1.29639i) q^{76} +(-0.239446 - 1.66538i) q^{79} +(0.654861 - 0.755750i) q^{80} +(0.415415 + 0.909632i) q^{81} +(-0.304632 + 0.474017i) q^{83} +(1.49611 + 0.215109i) q^{85} +(-0.654861 + 0.755750i) q^{90} +(0.841254 - 0.540641i) q^{92} -1.97964i q^{93} +(1.37491 + 1.19136i) q^{94} +(0.557730 - 1.89945i) q^{95} +(0.654861 - 0.755750i) q^{96} +(-0.841254 + 0.540641i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{8} - q^{9} - q^{10} - q^{12} + q^{15} - q^{16} - 11 q^{17} + q^{18} + q^{20} - q^{23} + q^{24} - q^{25} - q^{27} - q^{30} + q^{32} - q^{36} + 10 q^{40} - 10 q^{45} + q^{46} + 10 q^{48} - q^{49} + q^{50} - 2 q^{53} - 10 q^{54} + q^{60} + 2 q^{61} - q^{64} - q^{69} + q^{72} - q^{75} - 2 q^{79} + q^{80} - q^{81} - q^{90} - q^{92} + q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(e\left(\frac{7}{22}\right)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.654861 0.755750i 0.654861 0.755750i
\(3\) −0.959493 0.281733i −0.959493 0.281733i
\(4\) −0.142315 0.989821i −0.142315 0.989821i
\(5\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(6\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(7\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(8\) −0.841254 0.540641i −0.841254 0.540641i
\(9\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(10\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(11\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(12\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(13\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(14\) 0 0
\(15\) 0.959493 0.281733i 0.959493 0.281733i
\(16\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(17\) −1.14231 0.989821i −1.14231 0.989821i −0.142315 0.989821i \(-0.545455\pi\)
−1.00000 \(\pi\)
\(18\) 0.959493 0.281733i 0.959493 0.281733i
\(19\) −1.49611 + 1.29639i −1.49611 + 1.29639i −0.654861 + 0.755750i \(0.727273\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(20\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(21\) 0 0
\(22\) 0 0
\(23\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(24\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(25\) 0.415415 0.909632i 0.415415 0.909632i
\(26\) 0 0
\(27\) −0.654861 0.755750i −0.654861 0.755750i
\(28\) 0 0
\(29\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(30\) 0.415415 0.909632i 0.415415 0.909632i
\(31\) 0.557730 + 1.89945i 0.557730 + 1.89945i 0.415415 + 0.909632i \(0.363636\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(32\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(33\) 0 0
\(34\) −1.49611 + 0.215109i −1.49611 + 0.215109i
\(35\) 0 0
\(36\) 0.415415 0.909632i 0.415415 0.909632i
\(37\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(38\) 1.97964i 1.97964i
\(39\) 0 0
\(40\) 1.00000 1.00000
\(41\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(42\) 0 0
\(43\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(44\) 0 0
\(45\) −1.00000 −1.00000
\(46\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(47\) 1.81926i 1.81926i 0.415415 + 0.909632i \(0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(48\) 1.00000 1.00000
\(49\) −0.959493 0.281733i −0.959493 0.281733i
\(50\) −0.415415 0.909632i −0.415415 0.909632i
\(51\) 0.817178 + 1.27155i 0.817178 + 1.27155i
\(52\) 0 0
\(53\) −0.118239 + 0.822373i −0.118239 + 0.822373i 0.841254 + 0.540641i \(0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) 1.80075 0.822373i 1.80075 0.822373i
\(58\) 0 0
\(59\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(60\) −0.415415 0.909632i −0.415415 0.909632i
\(61\) 0.797176 0.234072i 0.797176 0.234072i 0.142315 0.989821i \(-0.454545\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(62\) 1.80075 + 0.822373i 1.80075 + 0.822373i
\(63\) 0 0
\(64\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(68\) −0.817178 + 1.27155i −0.817178 + 1.27155i
\(69\) −0.142315 0.989821i −0.142315 0.989821i
\(70\) 0 0
\(71\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(72\) −0.415415 0.909632i −0.415415 0.909632i
\(73\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(74\) 0 0
\(75\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(76\) 1.49611 + 1.29639i 1.49611 + 1.29639i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.239446 1.66538i −0.239446 1.66538i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(80\) 0.654861 0.755750i 0.654861 0.755750i
\(81\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(82\) 0 0
\(83\) −0.304632 + 0.474017i −0.304632 + 0.474017i −0.959493 0.281733i \(-0.909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(84\) 0 0
\(85\) 1.49611 + 0.215109i 1.49611 + 0.215109i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(90\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(91\) 0 0
\(92\) 0.841254 0.540641i 0.841254 0.540641i
\(93\) 1.97964i 1.97964i
\(94\) 1.37491 + 1.19136i 1.37491 + 1.19136i
\(95\) 0.557730 1.89945i 0.557730 1.89945i
\(96\) 0.654861 0.755750i 0.654861 0.755750i
\(97\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(98\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(99\) 0 0
\(100\) −0.959493 0.281733i −0.959493 0.281733i
\(101\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(102\) 1.49611 + 0.215109i 1.49611 + 0.215109i
\(103\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.544078 + 0.627899i 0.544078 + 0.627899i
\(107\) 0.304632 + 1.03748i 0.304632 + 1.03748i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(108\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(109\) −1.10181 + 1.27155i −1.10181 + 1.27155i −0.142315 + 0.989821i \(0.545455\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.512546 0.234072i −0.512546 0.234072i 0.142315 0.989821i \(-0.454545\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(114\) 0.557730 1.89945i 0.557730 1.89945i
\(115\) −0.841254 0.540641i −0.841254 0.540641i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −0.959493 0.281733i −0.959493 0.281733i
\(121\) 0.654861 0.755750i 0.654861 0.755750i
\(122\) 0.345139 0.755750i 0.345139 0.755750i
\(123\) 0 0
\(124\) 1.80075 0.822373i 1.80075 0.822373i
\(125\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(126\) 0 0
\(127\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(128\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(136\) 0.425839 + 1.45027i 0.425839 + 1.45027i
\(137\) 1.08128i 1.08128i 0.841254 + 0.540641i \(0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(138\) −0.841254 0.540641i −0.841254 0.540641i
\(139\) −0.830830 −0.830830 −0.415415 0.909632i \(-0.636364\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(140\) 0 0
\(141\) 0.512546 1.74557i 0.512546 1.74557i
\(142\) 0 0
\(143\) 0 0
\(144\) −0.959493 0.281733i −0.959493 0.281733i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(148\) 0 0
\(149\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(150\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(151\) −1.49611 + 0.215109i −1.49611 + 0.215109i −0.841254 0.540641i \(-0.818182\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(152\) 1.95949 0.281733i 1.95949 0.281733i
\(153\) −0.425839 1.45027i −0.425839 1.45027i
\(154\) 0 0
\(155\) −1.49611 1.29639i −1.49611 1.29639i
\(156\) 0 0
\(157\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(158\) −1.41542 0.909632i −1.41542 0.909632i
\(159\) 0.345139 0.755750i 0.345139 0.755750i
\(160\) −0.142315 0.989821i −0.142315 0.989821i
\(161\) 0 0
\(162\) 0.959493 + 0.281733i 0.959493 + 0.281733i
\(163\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.158746 + 0.540641i 0.158746 + 0.540641i
\(167\) −0.817178 0.708089i −0.817178 0.708089i 0.142315 0.989821i \(-0.454545\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(168\) 0 0
\(169\) 0.959493 0.281733i 0.959493 0.281733i
\(170\) 1.14231 0.989821i 1.14231 0.989821i
\(171\) −1.95949 + 0.281733i −1.95949 + 0.281733i
\(172\) 0 0
\(173\) 1.65486 0.755750i 1.65486 0.755750i 0.654861 0.755750i \(-0.272727\pi\)
1.00000 \(0\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(180\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(181\) −1.25667 0.368991i −1.25667 0.368991i −0.415415 0.909632i \(-0.636364\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(182\) 0 0
\(183\) −0.830830 −0.830830
\(184\) 0.142315 0.989821i 0.142315 0.989821i
\(185\) 0 0
\(186\) −1.49611 1.29639i −1.49611 1.29639i
\(187\) 0 0
\(188\) 1.80075 0.258908i 1.80075 0.258908i
\(189\) 0 0
\(190\) −1.07028 1.66538i −1.07028 1.66538i
\(191\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(192\) −0.142315 0.989821i −0.142315 0.989821i
\(193\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(197\) −1.07028 + 0.153882i −1.07028 + 0.153882i −0.654861 0.755750i \(-0.727273\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(198\) 0 0
\(199\) −1.84125 + 0.540641i −1.84125 + 0.540641i −0.841254 + 0.540641i \(0.818182\pi\)
−1.00000 \(1.00000\pi\)
\(200\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 1.14231 0.989821i 1.14231 0.989821i
\(205\) 0 0
\(206\) 0 0
\(207\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.857685 0.989821i −0.857685 0.989821i 0.142315 0.989821i \(-0.454545\pi\)
−1.00000 \(\pi\)
\(212\) 0.830830 0.830830
\(213\) 0 0
\(214\) 0.983568 + 0.449181i 0.983568 + 0.449181i
\(215\) 0 0
\(216\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(217\) 0 0
\(218\) 0.239446 + 1.66538i 0.239446 + 1.66538i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(224\) 0 0
\(225\) 0.841254 0.540641i 0.841254 0.540641i
\(226\) −0.512546 + 0.234072i −0.512546 + 0.234072i
\(227\) 0.512546 1.74557i 0.512546 1.74557i −0.142315 0.989821i \(-0.545455\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(228\) −1.07028 1.66538i −1.07028 1.66538i
\(229\) −1.91899 −1.91899 −0.959493 0.281733i \(-0.909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(230\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(231\) 0 0
\(232\) 0 0
\(233\) −1.61435 0.474017i −1.61435 0.474017i −0.654861 0.755750i \(-0.727273\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(234\) 0 0
\(235\) −0.983568 1.53046i −0.983568 1.53046i
\(236\) 0 0
\(237\) −0.239446 + 1.66538i −0.239446 + 1.66538i
\(238\) 0 0
\(239\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(240\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(241\) −1.37491 + 0.627899i −1.37491 + 0.627899i −0.959493 0.281733i \(-0.909091\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(242\) −0.142315 0.989821i −0.142315 0.989821i
\(243\) −0.142315 0.989821i −0.142315 0.989821i
\(244\) −0.345139 0.755750i −0.345139 0.755750i
\(245\) 0.959493 0.281733i 0.959493 0.281733i
\(246\) 0 0
\(247\) 0 0
\(248\) 0.557730 1.89945i 0.557730 1.89945i
\(249\) 0.425839 0.368991i 0.425839 0.368991i
\(250\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(251\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.37491 0.627899i −1.37491 0.627899i
\(256\) 0.841254 0.540641i 0.841254 0.540641i
\(257\) 0.857685 + 0.989821i 0.857685 + 0.989821i 1.00000 \(0\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.0405070 0.281733i −0.0405070 0.281733i 0.959493 0.281733i \(-0.0909091\pi\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −0.345139 0.755750i −0.345139 0.755750i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(270\) 0.841254 0.540641i 0.841254 0.540641i
\(271\) 0.983568 + 1.53046i 0.983568 + 1.53046i 0.841254 + 0.540641i \(0.181818\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(272\) 1.37491 + 0.627899i 1.37491 + 0.627899i
\(273\) 0 0
\(274\) 0.817178 + 0.708089i 0.817178 + 0.708089i
\(275\) 0 0
\(276\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −0.544078 + 0.627899i −0.544078 + 0.627899i
\(279\) −0.557730 + 1.89945i −0.557730 + 1.89945i
\(280\) 0 0
\(281\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(282\) −0.983568 1.53046i −0.983568 1.53046i
\(283\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(284\) 0 0
\(285\) −1.07028 + 1.66538i −1.07028 + 1.66538i
\(286\) 0 0
\(287\) 0 0
\(288\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(289\) 0.182822 + 1.27155i 0.182822 + 1.27155i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.25667 1.45027i 1.25667 1.45027i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(294\) 0.959493 0.281733i 0.959493 0.281733i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(301\) 0 0
\(302\) −0.817178 + 1.27155i −0.817178 + 1.27155i
\(303\) 0 0
\(304\) 1.07028 1.66538i 1.07028 1.66538i
\(305\) −0.544078 + 0.627899i −0.544078 + 0.627899i
\(306\) −1.37491 0.627899i −1.37491 0.627899i
\(307\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.95949 + 0.281733i −1.95949 + 0.281733i
\(311\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(312\) 0 0
\(313\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.61435 + 0.474017i −1.61435 + 0.474017i
\(317\) 0.817178 + 1.27155i 0.817178 + 1.27155i 0.959493 + 0.281733i \(0.0909091\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(318\) −0.345139 0.755750i −0.345139 0.755750i
\(319\) 0 0
\(320\) −0.841254 0.540641i −0.841254 0.540641i
\(321\) 1.08128i 1.08128i
\(322\) 0 0
\(323\) 2.99223 2.99223
\(324\) 0.841254 0.540641i 0.841254 0.540641i
\(325\) 0 0
\(326\) 0 0
\(327\) 1.41542 0.909632i 1.41542 0.909632i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.61435 + 1.03748i 1.61435 + 1.03748i 0.959493 + 0.281733i \(0.0909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(332\) 0.512546 + 0.234072i 0.512546 + 0.234072i
\(333\) 0 0
\(334\) −1.07028 + 0.153882i −1.07028 + 0.153882i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(338\) 0.415415 0.909632i 0.415415 0.909632i
\(339\) 0.425839 + 0.368991i 0.425839 + 0.368991i
\(340\) 1.51150i 1.51150i
\(341\) 0 0
\(342\) −1.07028 + 1.66538i −1.07028 + 1.66538i
\(343\) 0 0
\(344\) 0 0
\(345\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(346\) 0.512546 1.74557i 0.512546 1.74557i
\(347\) 0.118239 0.258908i 0.118239 0.258908i −0.841254 0.540641i \(-0.818182\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(348\) 0 0
\(349\) 0.425839 0.368991i 0.425839 0.368991i −0.415415 0.909632i \(-0.636364\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.273100 + 0.0801894i −0.273100 + 0.0801894i −0.415415 0.909632i \(-0.636364\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(360\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(361\) 0.415415 2.88927i 0.415415 2.88927i
\(362\) −1.10181 + 0.708089i −1.10181 + 0.708089i
\(363\) −0.841254 + 0.540641i −0.841254 + 0.540641i
\(364\) 0 0
\(365\) 0 0
\(366\) −0.544078 + 0.627899i −0.544078 + 0.627899i
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −0.654861 0.755750i −0.654861 0.755750i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.95949 + 0.281733i −1.95949 + 0.281733i
\(373\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(374\) 0 0
\(375\) 0.142315 0.989821i 0.142315 0.989821i
\(376\) 0.983568 1.53046i 0.983568 1.53046i
\(377\) 0 0
\(378\) 0 0
\(379\) −0.512546 + 0.234072i −0.512546 + 0.234072i −0.654861 0.755750i \(-0.727273\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(380\) −1.95949 0.281733i −1.95949 0.281733i
\(381\) 0 0
\(382\) 0 0
\(383\) −1.84125 + 0.540641i −1.84125 + 0.540641i −0.841254 + 0.540641i \(0.818182\pi\)
−1.00000 \(1.00000\pi\)
\(384\) −0.841254 0.540641i −0.841254 0.540641i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(390\) 0 0
\(391\) 0.425839 1.45027i 0.425839 1.45027i
\(392\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(393\) 0 0
\(394\) −0.584585 + 0.909632i −0.584585 + 0.909632i
\(395\) 1.10181 + 1.27155i 1.10181 + 1.27155i
\(396\) 0 0
\(397\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(398\) −0.797176 + 1.74557i −0.797176 + 1.74557i
\(399\) 0 0
\(400\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(401\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.841254 0.540641i −0.841254 0.540641i
\(406\) 0 0
\(407\) 0 0
\(408\) 1.51150i 1.51150i
\(409\) 1.10181 0.708089i 1.10181 0.708089i 0.142315 0.989821i \(-0.454545\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(410\) 0 0
\(411\) 0.304632 1.03748i 0.304632 1.03748i
\(412\) 0 0
\(413\) 0 0
\(414\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(415\) 0.563465i 0.563465i
\(416\) 0 0
\(417\) 0.797176 + 0.234072i 0.797176 + 0.234072i
\(418\) 0 0
\(419\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(420\) 0 0
\(421\) −0.0405070 + 0.281733i −0.0405070 + 0.281733i 0.959493 + 0.281733i \(0.0909091\pi\)
−1.00000 \(1.00000\pi\)
\(422\) −1.30972 −1.30972
\(423\) −0.983568 + 1.53046i −0.983568 + 1.53046i
\(424\) 0.544078 0.627899i 0.544078 0.627899i
\(425\) −1.37491 + 0.627899i −1.37491 + 0.627899i
\(426\) 0 0
\(427\) 0 0
\(428\) 0.983568 0.449181i 0.983568 0.449181i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(432\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(433\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.41542 + 0.909632i 1.41542 + 0.909632i
\(437\) −1.80075 0.822373i −1.80075 0.822373i
\(438\) 0 0
\(439\) 0.512546 + 0.234072i 0.512546 + 0.234072i 0.654861 0.755750i \(-0.272727\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(440\) 0 0
\(441\) −0.654861 0.755750i −0.654861 0.755750i
\(442\) 0 0
\(443\) −0.857685 + 0.989821i −0.857685 + 0.989821i 0.142315 + 0.989821i \(0.454545\pi\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(450\) 0.142315 0.989821i 0.142315 0.989821i
\(451\) 0 0
\(452\) −0.158746 + 0.540641i −0.158746 + 0.540641i
\(453\) 1.49611 + 0.215109i 1.49611 + 0.215109i
\(454\) −0.983568 1.53046i −0.983568 1.53046i
\(455\) 0 0
\(456\) −1.95949 0.281733i −1.95949 0.281733i
\(457\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(458\) −1.25667 + 1.45027i −1.25667 + 1.45027i
\(459\) 1.51150i 1.51150i
\(460\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(464\) 0 0
\(465\) 1.07028 + 1.66538i 1.07028 + 1.66538i
\(466\) −1.41542 + 0.909632i −1.41542 + 0.909632i
\(467\) 1.49611 + 0.215109i 1.49611 + 0.215109i 0.841254 0.540641i \(-0.181818\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.80075 0.258908i −1.80075 0.258908i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 1.10181 + 1.27155i 1.10181 + 1.27155i
\(475\) 0.557730 + 1.89945i 0.557730 + 1.89945i
\(476\) 0 0
\(477\) −0.544078 + 0.627899i −0.544078 + 0.627899i
\(478\) 0 0
\(479\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(480\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(481\) 0 0
\(482\) −0.425839 + 1.45027i −0.425839 + 1.45027i
\(483\) 0 0
\(484\) −0.841254 0.540641i −0.841254 0.540641i
\(485\) 0 0
\(486\) −0.841254 0.540641i −0.841254 0.540641i
\(487\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(488\) −0.797176 0.234072i −0.797176 0.234072i
\(489\) 0 0
\(490\) 0.415415 0.909632i 0.415415 0.909632i
\(491\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.07028 1.66538i −1.07028 1.66538i
\(497\) 0 0
\(498\) 0.563465i 0.563465i
\(499\) 0.239446 1.66538i 0.239446 1.66538i −0.415415 0.909632i \(-0.636364\pi\)
0.654861 0.755750i \(-0.272727\pi\)
\(500\) 0.959493 0.281733i 0.959493 0.281733i
\(501\) 0.584585 + 0.909632i 0.584585 + 0.909632i
\(502\) 0 0
\(503\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i 0.415415 0.909632i \(-0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −1.00000
\(508\) 0 0
\(509\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(510\) −1.37491 + 0.627899i −1.37491 + 0.627899i
\(511\) 0 0
\(512\) 0.142315 0.989821i 0.142315 0.989821i
\(513\) 1.95949 + 0.281733i 1.95949 + 0.281733i
\(514\) 1.30972 1.30972
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.80075 + 0.258908i −1.80075 + 0.258908i
\(520\) 0 0
\(521\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(522\) 0 0
\(523\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.239446 0.153882i −0.239446 0.153882i
\(527\) 1.24302 2.72183i 1.24302 2.72183i
\(528\) 0 0
\(529\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(530\) −0.797176 0.234072i −0.797176 0.234072i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.817178 0.708089i −0.817178 0.708089i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0.142315 0.989821i 0.142315 0.989821i
\(541\) 1.80075 0.822373i 1.80075 0.822373i 0.841254 0.540641i \(-0.181818\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(542\) 1.80075 + 0.258908i 1.80075 + 0.258908i
\(543\) 1.10181 + 0.708089i 1.10181 + 0.708089i
\(544\) 1.37491 0.627899i 1.37491 0.627899i
\(545\) 0.239446 1.66538i 0.239446 1.66538i
\(546\) 0 0
\(547\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(548\) 1.07028 0.153882i 1.07028 0.153882i
\(549\) 0.797176 + 0.234072i 0.797176 + 0.234072i
\(550\) 0 0
\(551\) 0 0
\(552\) −0.415415 + 0.909632i −0.415415 + 0.909632i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(557\) 0.239446 0.153882i 0.239446 0.153882i −0.415415 0.909632i \(-0.636364\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(558\) 1.07028 + 1.66538i 1.07028 + 1.66538i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.65486 0.755750i 1.65486 0.755750i 0.654861 0.755750i \(-0.272727\pi\)
1.00000 \(0\)
\(564\) −1.80075 0.258908i −1.80075 0.258908i
\(565\) 0.557730 0.0801894i 0.557730 0.0801894i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(570\) 0.557730 + 1.89945i 0.557730 + 1.89945i
\(571\) 1.37491 1.19136i 1.37491 1.19136i 0.415415 0.909632i \(-0.363636\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 1.00000
\(576\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(577\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(578\) 1.08070 + 0.694523i 1.08070 + 0.694523i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −0.273100 1.89945i −0.273100 1.89945i
\(587\) 0.118239 + 0.258908i 0.118239 + 0.258908i 0.959493 0.281733i \(-0.0909091\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(588\) 0.415415 0.909632i 0.415415 0.909632i
\(589\) −3.29686 2.11876i −3.29686 2.11876i
\(590\) 0 0
\(591\) 1.07028 + 0.153882i 1.07028 + 0.153882i
\(592\) 0 0
\(593\) −0.239446 + 0.153882i −0.239446 + 0.153882i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.91899 1.91899
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0.959493 0.281733i 0.959493 0.281733i
\(601\) 1.61435 + 0.474017i 1.61435 + 0.474017i 0.959493 0.281733i \(-0.0909091\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0.425839 + 1.45027i 0.425839 + 1.45027i
\(605\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(606\) 0 0
\(607\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(608\) −0.557730 1.89945i −0.557730 1.89945i
\(609\) 0 0
\(610\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(611\) 0 0
\(612\) −1.37491 + 0.627899i −1.37491 + 0.627899i
\(613\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.37491 + 1.19136i −1.37491 + 1.19136i −0.415415 + 0.909632i \(0.636364\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(618\) 0 0
\(619\) −0.983568 0.449181i −0.983568 0.449181i −0.142315 0.989821i \(-0.545455\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(620\) −1.07028 + 1.66538i −1.07028 + 1.66538i
\(621\) 0.415415 0.909632i 0.415415 0.909632i
\(622\) 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
\(631\) 0.186393 + 1.29639i 0.186393 + 1.29639i 0.841254 + 0.540641i \(0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(632\) −0.698939 + 1.53046i −0.698939 + 1.53046i
\(633\) 0.544078 + 1.19136i 0.544078 + 1.19136i
\(634\) 1.49611 + 0.215109i 1.49611 + 0.215109i
\(635\) 0 0
\(636\) −0.797176 0.234072i −0.797176 0.234072i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(641\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(642\) −0.817178 0.708089i −0.817178 0.708089i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.95949 2.26138i 1.95949 2.26138i
\(647\) 0.425839 1.45027i 0.425839 1.45027i −0.415415 0.909632i \(-0.636364\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(648\) 0.142315 0.989821i 0.142315 0.989821i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.584585 0.909632i 0.584585 0.909632i −0.415415 0.909632i \(-0.636364\pi\)
1.00000 \(0\)
\(654\) 0.239446 1.66538i 0.239446 1.66538i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(660\) 0 0
\(661\) 0.186393 0.215109i 0.186393 0.215109i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(662\) 1.84125 0.540641i 1.84125 0.540641i
\(663\) 0 0
\(664\) 0.512546 0.234072i 0.512546 0.234072i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.584585 + 0.909632i −0.584585 + 0.909632i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(674\) 0 0
\(675\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(676\) −0.415415 0.909632i −0.415415 0.909632i
\(677\) 0.239446 + 1.66538i 0.239446 + 1.66538i 0.654861 + 0.755750i \(0.272727\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(678\) 0.557730 0.0801894i 0.557730 0.0801894i
\(679\) 0 0
\(680\) −1.14231 0.989821i −1.14231 0.989821i
\(681\) −0.983568 + 1.53046i −0.983568 + 1.53046i
\(682\) 0 0
\(683\) 0.118239 0.822373i 0.118239 0.822373i −0.841254 0.540641i \(-0.818182\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(684\) 0.557730 + 1.89945i 0.557730 + 1.89945i
\(685\) −0.584585 0.909632i −0.584585 0.909632i
\(686\) 0 0
\(687\) 1.84125 + 0.540641i 1.84125 + 0.540641i
\(688\) 0 0
\(689\) 0 0
\(690\) 1.00000 1.00000
\(691\) −1.91899 −1.91899 −0.959493 0.281733i \(-0.909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(692\) −0.983568 1.53046i −0.983568 1.53046i
\(693\) 0 0
\(694\) −0.118239 0.258908i −0.118239 0.258908i
\(695\) 0.698939 0.449181i 0.698939 0.449181i
\(696\) 0 0
\(697\) 0 0
\(698\) 0.563465i 0.563465i
\(699\) 1.41542 + 0.909632i 1.41542 + 0.909632i
\(700\) 0 0
\(701\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0.512546 + 1.74557i 0.512546 + 1.74557i
\(706\) −0.118239 + 0.258908i −0.118239 + 0.258908i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.857685 + 0.989821i 0.857685 + 0.989821i 1.00000 \(0\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(710\) 0 0
\(711\) 0.698939 1.53046i 0.698939 1.53046i
\(712\) 0 0
\(713\) −1.49611 + 1.29639i −1.49611 + 1.29639i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(720\) 0.959493 0.281733i 0.959493 0.281733i
\(721\) 0 0
\(722\) −1.91153 2.20602i −1.91153 2.20602i
\(723\) 1.49611 0.215109i 1.49611 0.215109i
\(724\) −0.186393 + 1.29639i −0.186393 + 1.29639i
\(725\) 0 0
\(726\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(727\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(728\) 0 0
\(729\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(730\) 0 0
\(731\) 0 0
\(732\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(733\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(734\) 0 0
\(735\) −1.00000 −1.00000
\(736\) −1.00000 −1.00000
\(737\) 0 0
\(738\) 0 0
\(739\) 1.84125 + 0.540641i 1.84125 + 0.540641i 1.00000 \(0\)
0.841254 + 0.540641i \(0.181818\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.273100 + 1.89945i −0.273100 + 1.89945i 0.142315 + 0.989821i \(0.454545\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(744\) −1.07028 + 1.66538i −1.07028 + 1.66538i
\(745\) 0 0
\(746\) 0 0
\(747\) −0.512546 + 0.234072i −0.512546 + 0.234072i
\(748\) 0 0
\(749\) 0 0
\(750\) −0.654861 0.755750i −0.654861 0.755750i
\(751\) −1.25667 + 0.368991i −1.25667 + 0.368991i −0.841254 0.540641i \(-0.818182\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(752\) −0.512546 1.74557i −0.512546 1.74557i
\(753\) 0 0
\(754\) 0 0
\(755\) 1.14231 0.989821i 1.14231 0.989821i
\(756\) 0 0
\(757\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(758\) −0.158746 + 0.540641i −0.158746 + 0.540641i
\(759\) 0 0
\(760\) −1.49611 + 1.29639i −1.49611 + 1.29639i
\(761\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.14231 + 0.989821i 1.14231 + 0.989821i
\(766\) −0.797176 + 1.74557i −0.797176 + 1.74557i
\(767\) 0 0
\(768\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(769\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(770\) 0 0
\(771\) −0.544078 1.19136i −0.544078 1.19136i
\(772\) 0 0
\(773\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(774\) 0 0
\(775\) 1.95949 + 0.281733i 1.95949 + 0.281733i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −0.817178 1.27155i −0.817178 1.27155i
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(788\) 0.304632 + 1.03748i 0.304632 + 1.03748i
\(789\) −0.0405070 + 0.281733i −0.0405070 + 0.281733i
\(790\) 1.68251 1.68251
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(796\) 0.797176 + 1.74557i 0.797176 + 1.74557i
\(797\) −0.273100 + 0.0801894i −0.273100 + 0.0801894i −0.415415 0.909632i \(-0.636364\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(798\) 0 0
\(799\) 1.80075 2.07817i 1.80075 2.07817i
\(800\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(810\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(811\) −1.10181 + 1.27155i −1.10181 + 1.27155i −0.142315 + 0.989821i \(0.545455\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(812\) 0 0
\(813\) −0.512546 1.74557i −0.512546 1.74557i
\(814\) 0 0
\(815\) 0 0
\(816\) −1.14231 0.989821i −1.14231 0.989821i
\(817\) 0 0
\(818\) 0.186393 1.29639i 0.186393 1.29639i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(822\) −0.584585 0.909632i −0.584585 0.909632i
\(823\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.97964i 1.97964i 0.142315 + 0.989821i \(0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(828\) 1.00000 1.00000
\(829\) 1.08128i 1.08128i 0.841254 + 0.540641i \(0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(830\) −0.425839 0.368991i −0.425839 0.368991i
\(831\) 0 0
\(832\) 0 0
\(833\) 0.817178 + 1.27155i 0.817178 + 1.27155i
\(834\) 0.698939 0.449181i 0.698939 0.449181i
\(835\) 1.07028 + 0.153882i 1.07028 + 0.153882i
\(836\) 0 0
\(837\) 1.07028 1.66538i 1.07028 1.66538i
\(838\) 0 0
\(839\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(840\) 0 0
\(841\) 0.142315 + 0.989821i 0.142315 + 0.989821i
\(842\) 0.186393 + 0.215109i 0.186393 + 0.215109i
\(843\) 0 0
\(844\) −0.857685 + 0.989821i −0.857685 + 0.989821i
\(845\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(846\) 0.512546 + 1.74557i 0.512546 + 1.74557i
\(847\) 0 0
\(848\) −0.118239 0.822373i −0.118239 0.822373i
\(849\) 0 0
\(850\) −0.425839 + 1.45027i −0.425839 + 1.45027i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(854\) 0 0
\(855\) 1.49611 1.29639i 1.49611 1.29639i
\(856\) 0.304632 1.03748i 0.304632 1.03748i
\(857\) −0.186393 + 0.215109i −0.186393 + 0.215109i −0.841254 0.540641i \(-0.818182\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(858\) 0 0
\(859\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.983568 + 0.449181i −0.983568 + 0.449181i −0.841254 0.540641i \(-0.818182\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(864\) 0.959493 0.281733i 0.959493 0.281733i
\(865\) −0.983568 + 1.53046i −0.983568 + 1.53046i
\(866\) 0 0
\(867\) 0.182822 1.27155i 0.182822 1.27155i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.61435 0.474017i 1.61435 0.474017i
\(873\) 0 0
\(874\) −1.80075 + 0.822373i −1.80075 + 0.822373i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(878\) 0.512546 0.234072i 0.512546 0.234072i
\(879\) −1.61435 + 1.03748i −1.61435 + 1.03748i
\(880\) 0 0
\(881\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(882\) −1.00000 −1.00000
\(883\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.186393 + 1.29639i 0.186393 + 1.29639i
\(887\) −1.49611 + 0.215109i −1.49611 + 0.215109i −0.841254 0.540641i \(-0.818182\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.35848 2.72183i −2.35848 2.72183i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.654861 0.755750i −0.654861 0.755750i
\(901\) 0.949069 0.822373i 0.949069 0.822373i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.304632 + 0.474017i 0.304632 + 0.474017i
\(905\) 1.25667 0.368991i 1.25667 0.368991i
\(906\) 1.14231 0.989821i 1.14231 0.989821i
\(907\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(908\) −1.80075 0.258908i −1.80075 0.258908i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(912\) −1.49611 + 1.29639i −1.49611 + 1.29639i
\(913\) 0 0
\(914\) 0 0
\(915\) 0.698939 0.449181i 0.698939 0.449181i
\(916\) 0.273100 + 1.89945i 0.273100 + 1.89945i
\(917\) 0 0
\(918\) 1.14231 + 0.989821i 1.14231 + 0.989821i
\(919\) 1.30972 1.30972 0.654861 0.755750i \(-0.272727\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(920\) 0.415415 + 0.909632i 0.415415 + 0.909632i
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(930\) 1.95949 + 0.281733i 1.95949 + 0.281733i
\(931\) 1.80075 0.822373i 1.80075 0.822373i
\(932\) −0.239446 + 1.66538i −0.239446 + 1.66538i
\(933\) 0 0
\(934\) 1.14231 0.989821i 1.14231 0.989821i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.37491 + 1.19136i −1.37491 + 1.19136i
\(941\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.544078 0.627899i −0.544078 0.627899i 0.415415 0.909632i \(-0.363636\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(948\) 1.68251 1.68251
\(949\) 0 0
\(950\) 1.80075 + 0.822373i 1.80075 + 0.822373i
\(951\) −0.425839 1.45027i −0.425839 1.45027i
\(952\) 0 0
\(953\) 1.80075 0.258908i 1.80075 0.258908i 0.841254 0.540641i \(-0.181818\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(954\) 0.118239 + 0.822373i 0.118239 + 0.822373i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0.654861 + 0.755750i 0.654861 + 0.755750i
\(961\) −2.45561 + 1.57812i −2.45561 + 1.57812i
\(962\) 0 0
\(963\) −0.304632 + 1.03748i −0.304632 + 1.03748i
\(964\) 0.817178 + 1.27155i 0.817178 + 1.27155i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(969\) −2.87102 0.843008i −2.87102 0.843008i
\(970\) 0 0
\(971\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(972\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −0.698939 + 0.449181i −0.698939 + 0.449181i
\(977\) 0.983568 0.449181i 0.983568 0.449181i 0.142315 0.989821i \(-0.454545\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.415415 0.909632i −0.415415 0.909632i
\(981\) −1.61435 + 0.474017i −1.61435 + 0.474017i
\(982\) 0 0
\(983\) 1.10181 1.27155i 1.10181 1.27155i 0.142315 0.989821i \(-0.454545\pi\)
0.959493 0.281733i \(-0.0909091\pi\)
\(984\) 0 0
\(985\) 0.817178 0.708089i 0.817178 0.708089i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.983568 0.449181i −0.983568 0.449181i −0.142315 0.989821i \(-0.545455\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(992\) −1.95949 0.281733i −1.95949 0.281733i
\(993\) −1.25667 1.45027i −1.25667 1.45027i
\(994\) 0 0
\(995\) 1.25667 1.45027i 1.25667 1.45027i
\(996\) −0.425839 0.368991i −0.425839 0.368991i
\(997\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(998\) −1.10181 1.27155i −1.10181 1.27155i
\(999\) 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 2760.1.co.d.1259.1 yes 10
3.2 odd 2 2760.1.co.a.1259.1 10
5.4 even 2 2760.1.co.a.1259.1 10
8.3 odd 2 2760.1.co.c.1259.1 yes 10
15.14 odd 2 CM 2760.1.co.d.1259.1 yes 10
23.19 odd 22 2760.1.co.c.1859.1 yes 10
24.11 even 2 2760.1.co.b.1259.1 yes 10
40.19 odd 2 2760.1.co.b.1259.1 yes 10
69.65 even 22 2760.1.co.b.1859.1 yes 10
115.19 odd 22 2760.1.co.b.1859.1 yes 10
120.59 even 2 2760.1.co.c.1259.1 yes 10
184.19 even 22 inner 2760.1.co.d.1859.1 yes 10
345.134 even 22 2760.1.co.c.1859.1 yes 10
552.203 odd 22 2760.1.co.a.1859.1 yes 10
920.19 even 22 2760.1.co.a.1859.1 yes 10
2760.1859 odd 22 inner 2760.1.co.d.1859.1 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.1.co.a.1259.1 10 3.2 odd 2
2760.1.co.a.1259.1 10 5.4 even 2
2760.1.co.a.1859.1 yes 10 552.203 odd 22
2760.1.co.a.1859.1 yes 10 920.19 even 22
2760.1.co.b.1259.1 yes 10 24.11 even 2
2760.1.co.b.1259.1 yes 10 40.19 odd 2
2760.1.co.b.1859.1 yes 10 69.65 even 22
2760.1.co.b.1859.1 yes 10 115.19 odd 22
2760.1.co.c.1259.1 yes 10 8.3 odd 2
2760.1.co.c.1259.1 yes 10 120.59 even 2
2760.1.co.c.1859.1 yes 10 23.19 odd 22
2760.1.co.c.1859.1 yes 10 345.134 even 22
2760.1.co.d.1259.1 yes 10 1.1 even 1 trivial
2760.1.co.d.1259.1 yes 10 15.14 odd 2 CM
2760.1.co.d.1859.1 yes 10 184.19 even 22 inner
2760.1.co.d.1859.1 yes 10 2760.1859 odd 22 inner