Properties

Label 2151.1.f.b.238.12
Level $2151$
Weight $1$
Character 2151.238
Analytic conductor $1.073$
Analytic rank $0$
Dimension $24$
Projective image $D_{45}$
CM discriminant -239
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2151,1,Mod(238,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2151.238");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2151.f (of order \(6\), degree \(2\), 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: \(1.07348884217\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{45})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{45}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\)

Embedding invariants

Embedding label 238.12
Root \(-0.374607 + 0.927184i\) of defining polynomial
Character \(\chi\) \(=\) 2151.238
Dual form 2151.1.f.b.1672.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.997564 + 1.72783i) q^{2} +(0.913545 - 0.406737i) q^{3} +(-1.49027 + 2.58122i) q^{4} +(-0.961262 + 1.66495i) q^{5} +(1.61409 + 1.17271i) q^{6} -3.95142 q^{8} +(0.669131 - 0.743145i) q^{9} +O(q^{10})\) \(q+(0.997564 + 1.72783i) q^{2} +(0.913545 - 0.406737i) q^{3} +(-1.49027 + 2.58122i) q^{4} +(-0.961262 + 1.66495i) q^{5} +(1.61409 + 1.17271i) q^{6} -3.95142 q^{8} +(0.669131 - 0.743145i) q^{9} -3.83568 q^{10} +(-0.0348995 - 0.0604477i) q^{11} +(-0.311551 + 2.96421i) q^{12} +(-0.200958 + 1.91199i) q^{15} +(-2.45153 - 4.24617i) q^{16} +1.53209 q^{17} +(1.95153 + 0.414810i) q^{18} +(-2.86508 - 4.96246i) q^{20} +(0.0696290 - 0.120601i) q^{22} +(-3.60980 + 1.60719i) q^{24} +(-1.34805 - 2.33489i) q^{25} +(0.309017 - 0.951057i) q^{27} +(0.615661 + 1.06636i) q^{29} +(-3.50407 + 1.56011i) q^{30} +(-0.438371 + 0.759281i) q^{31} +(2.91540 - 5.04963i) q^{32} +(-0.0564686 - 0.0410268i) q^{33} +(1.52836 + 2.64719i) q^{34} +(0.921036 + 2.83466i) q^{36} +(3.79835 - 6.57894i) q^{40} +0.208038 q^{44} +(0.594092 + 1.82843i) q^{45} +(-3.96666 - 2.88195i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(2.68953 - 4.65840i) q^{50} +(1.39963 - 0.623157i) q^{51} +(1.95153 - 0.414810i) q^{54} +0.134190 q^{55} +(-1.22832 + 2.12752i) q^{58} +(-4.63579 - 3.36810i) q^{60} +(-0.766044 - 1.32683i) q^{61} -1.74921 q^{62} +6.73015 q^{64} +(0.0145564 - 0.138495i) q^{66} +(-0.173648 + 0.300767i) q^{67} +(-2.28322 + 3.95466i) q^{68} +0.618034 q^{71} +(-2.64402 + 2.93648i) q^{72} +(-2.18119 - 1.58473i) q^{75} +9.42625 q^{80} +(-0.104528 - 0.994522i) q^{81} +(0.374607 + 0.648838i) q^{83} +(-1.47274 + 2.55086i) q^{85} +(0.996161 + 0.723753i) q^{87} +(0.137903 + 0.238854i) q^{88} +(-2.56657 + 2.85047i) q^{90} +(-0.0916445 + 0.871939i) q^{93} +(0.609485 - 5.79887i) q^{96} -1.99513 q^{98} +(-0.0682737 - 0.0145120i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9} + 6 q^{10} + 3 q^{12} - 12 q^{16} - 3 q^{20} - 3 q^{22} - 3 q^{24} - 12 q^{25} - 6 q^{27} - 3 q^{30} - 3 q^{32} + 6 q^{34} - 6 q^{36} - 3 q^{40} + 6 q^{44} - 6 q^{48} - 12 q^{49} + 6 q^{50} - 12 q^{55} - 3 q^{58} - 9 q^{60} - 24 q^{62} + 30 q^{64} + 27 q^{66} - 3 q^{68} - 12 q^{71} - 18 q^{72} - 6 q^{75} + 54 q^{80} + 3 q^{81} - 3 q^{85} - 3 q^{88} - 18 q^{90} - 3 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times\).

\(n\) \(479\) \(1441\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.997564 + 1.72783i 0.997564 + 1.72783i 0.559193 + 0.829038i \(0.311111\pi\)
0.438371 + 0.898794i \(0.355556\pi\)
\(3\) 0.913545 0.406737i 0.913545 0.406737i
\(4\) −1.49027 + 2.58122i −1.49027 + 2.58122i
\(5\) −0.961262 + 1.66495i −0.961262 + 1.66495i −0.241922 + 0.970296i \(0.577778\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(6\) 1.61409 + 1.17271i 1.61409 + 1.17271i
\(7\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) −3.95142 −3.95142
\(9\) 0.669131 0.743145i 0.669131 0.743145i
\(10\) −3.83568 −3.83568
\(11\) −0.0348995 0.0604477i −0.0348995 0.0604477i 0.848048 0.529919i \(-0.177778\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(12\) −0.311551 + 2.96421i −0.311551 + 2.96421i
\(13\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) −0.200958 + 1.91199i −0.200958 + 1.91199i
\(16\) −2.45153 4.24617i −2.45153 4.24617i
\(17\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(18\) 1.95153 + 0.414810i 1.95153 + 0.414810i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −2.86508 4.96246i −2.86508 4.96246i
\(21\) 0 0
\(22\) 0.0696290 0.120601i 0.0696290 0.120601i
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) −3.60980 + 1.60719i −3.60980 + 1.60719i
\(25\) −1.34805 2.33489i −1.34805 2.33489i
\(26\) 0 0
\(27\) 0.309017 0.951057i 0.309017 0.951057i
\(28\) 0 0
\(29\) 0.615661 + 1.06636i 0.615661 + 1.06636i 0.990268 + 0.139173i \(0.0444444\pi\)
−0.374607 + 0.927184i \(0.622222\pi\)
\(30\) −3.50407 + 1.56011i −3.50407 + 1.56011i
\(31\) −0.438371 + 0.759281i −0.438371 + 0.759281i −0.997564 0.0697565i \(-0.977778\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(32\) 2.91540 5.04963i 2.91540 5.04963i
\(33\) −0.0564686 0.0410268i −0.0564686 0.0410268i
\(34\) 1.52836 + 2.64719i 1.52836 + 2.64719i
\(35\) 0 0
\(36\) 0.921036 + 2.83466i 0.921036 + 2.83466i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 3.79835 6.57894i 3.79835 6.57894i
\(41\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0.208038 0.208038
\(45\) 0.594092 + 1.82843i 0.594092 + 1.82843i
\(46\) 0 0
\(47\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) −3.96666 2.88195i −3.96666 2.88195i
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 2.68953 4.65840i 2.68953 4.65840i
\(51\) 1.39963 0.623157i 1.39963 0.623157i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.95153 0.414810i 1.95153 0.414810i
\(55\) 0.134190 0.134190
\(56\) 0 0
\(57\) 0 0
\(58\) −1.22832 + 2.12752i −1.22832 + 2.12752i
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) −4.63579 3.36810i −4.63579 3.36810i
\(61\) −0.766044 1.32683i −0.766044 1.32683i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(62\) −1.74921 −1.74921
\(63\) 0 0
\(64\) 6.73015 6.73015
\(65\) 0 0
\(66\) 0.0145564 0.138495i 0.0145564 0.138495i
\(67\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(68\) −2.28322 + 3.95466i −2.28322 + 3.95466i
\(69\) 0 0
\(70\) 0 0
\(71\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(72\) −2.64402 + 2.93648i −2.64402 + 2.93648i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −2.18119 1.58473i −2.18119 1.58473i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 9.42625 9.42625
\(81\) −0.104528 0.994522i −0.104528 0.994522i
\(82\) 0 0
\(83\) 0.374607 + 0.648838i 0.374607 + 0.648838i 0.990268 0.139173i \(-0.0444444\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(84\) 0 0
\(85\) −1.47274 + 2.55086i −1.47274 + 2.55086i
\(86\) 0 0
\(87\) 0.996161 + 0.723753i 0.996161 + 0.723753i
\(88\) 0.137903 + 0.238854i 0.137903 + 0.238854i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) −2.56657 + 2.85047i −2.56657 + 2.85047i
\(91\) 0 0
\(92\) 0 0
\(93\) −0.0916445 + 0.871939i −0.0916445 + 0.871939i
\(94\) 0 0
\(95\) 0 0
\(96\) 0.609485 5.79887i 0.609485 5.79887i
\(97\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) −1.99513 −1.99513
\(99\) −0.0682737 0.0145120i −0.0682737 0.0145120i
\(100\) 8.03581 8.03581
\(101\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(102\) 2.47293 + 1.79669i 2.47293 + 1.79669i
\(103\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.99437 + 2.21497i 1.99437 + 2.21497i
\(109\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(110\) 0.133863 + 0.231858i 0.133863 + 0.231858i
\(111\) 0 0
\(112\) 0 0
\(113\) 0.104528 0.181049i 0.104528 0.181049i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.67000 −3.67000
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0.794072 7.55509i 0.794072 7.55509i
\(121\) 0.497564 0.861806i 0.497564 0.861806i
\(122\) 1.52836 2.64719i 1.52836 2.64719i
\(123\) 0 0
\(124\) −1.30658 2.26306i −1.30658 2.26306i
\(125\) 3.26078 3.26078
\(126\) 0 0
\(127\) −1.76590 −1.76590 −0.882948 0.469472i \(-0.844444\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(128\) 3.79835 + 6.57894i 3.79835 + 6.57894i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0.190053 0.0846168i 0.190053 0.0846168i
\(133\) 0 0
\(134\) −0.692901 −0.692901
\(135\) 1.28642 + 1.42871i 1.28642 + 1.42871i
\(136\) −6.05393 −6.05393
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) 0 0
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.616528 + 1.06786i 0.616528 + 1.06786i
\(143\) 0 0
\(144\) −4.79592 1.01940i −4.79592 1.01940i
\(145\) −2.36725 −2.36725
\(146\) 0 0
\(147\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(148\) 0 0
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0.562265 5.34959i 0.562265 5.34959i
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 1.02517 1.13856i 1.02517 1.13856i
\(154\) 0 0
\(155\) −0.842779 1.45974i −0.842779 1.45974i
\(156\) 0 0
\(157\) 0.615661 1.06636i 0.615661 1.06636i −0.374607 0.927184i \(-0.622222\pi\)
0.990268 0.139173i \(-0.0444444\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 5.60493 + 9.70803i 5.60493 + 9.70803i
\(161\) 0 0
\(162\) 1.61409 1.17271i 1.61409 1.17271i
\(163\) −1.43868 −1.43868 −0.719340 0.694658i \(-0.755556\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(164\) 0 0
\(165\) 0.122589 0.0545801i 0.122589 0.0545801i
\(166\) −0.747388 + 1.29451i −0.747388 + 1.29451i
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) −0.500000 0.866025i −0.500000 0.866025i
\(170\) −5.87660 −5.87660
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) −0.256790 + 2.44319i −0.256790 + 2.44319i
\(175\) 0 0
\(176\) −0.171114 + 0.296379i −0.171114 + 0.296379i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −5.60493 1.19137i −5.60493 1.19137i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −1.23949 0.900539i −1.23949 0.900539i
\(184\) 0 0
\(185\) 0 0
\(186\) −1.59799 + 0.711469i −1.59799 + 0.711469i
\(187\) −0.0534691 0.0926113i −0.0534691 0.0926113i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 6.14830 2.73740i 6.14830 2.73740i
\(193\) 0.719340 1.24593i 0.719340 1.24593i −0.241922 0.970296i \(-0.577778\pi\)
0.961262 0.275637i \(-0.0888889\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.49027 2.58122i −1.49027 2.58122i
\(197\) 1.69610 1.69610 0.848048 0.529919i \(-0.177778\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(198\) −0.0430331 0.132442i −0.0430331 0.132442i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 5.32671 + 9.22613i 5.32671 + 9.22613i
\(201\) −0.0363024 + 0.345394i −0.0363024 + 0.345394i
\(202\) −1.61409 + 2.79569i −1.61409 + 2.79569i
\(203\) 0 0
\(204\) −0.477324 + 4.54143i −0.477324 + 4.54143i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.374607 0.648838i 0.374607 0.648838i −0.615661 0.788011i \(-0.711111\pi\)
0.990268 + 0.139173i \(0.0444444\pi\)
\(212\) 0 0
\(213\) 0.564602 0.251377i 0.564602 0.251377i
\(214\) 0 0
\(215\) 0 0
\(216\) −1.22106 + 3.75803i −1.22106 + 3.75803i
\(217\) 0 0
\(218\) 1.33500 + 2.31229i 1.33500 + 2.31229i
\(219\) 0 0
\(220\) −0.199979 + 0.346374i −0.199979 + 0.346374i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0 0
\(225\) −2.63718 0.560550i −2.63718 0.560550i
\(226\) 0.417095 0.417095
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.43274 4.21363i −2.43274 4.21363i
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(240\) 8.61130 3.83400i 8.61130 3.83400i
\(241\) 0.104528 + 0.181049i 0.104528 + 0.181049i 0.913545 0.406737i \(-0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(242\) 1.98541 1.98541
\(243\) −0.500000 0.866025i −0.500000 0.866025i
\(244\) 4.56645 4.56645
\(245\) −0.961262 1.66495i −0.961262 1.66495i
\(246\) 0 0
\(247\) 0 0
\(248\) 1.73219 3.00024i 1.73219 3.00024i
\(249\) 0.606126 + 0.440376i 0.606126 + 0.440376i
\(250\) 3.25284 + 5.63409i 3.25284 + 5.63409i
\(251\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.76159 3.05117i −1.76159 3.05117i
\(255\) −0.307886 + 2.92934i −0.307886 + 2.92934i
\(256\) −4.21312 + 7.29734i −4.21312 + 7.29734i
\(257\) 0.882948 1.52931i 0.882948 1.52931i 0.0348995 0.999391i \(-0.488889\pi\)
0.848048 0.529919i \(-0.177778\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.20442 + 0.256006i 1.20442 + 0.256006i
\(262\) 0 0
\(263\) −0.990268 1.71519i −0.990268 1.71519i −0.615661 0.788011i \(-0.711111\pi\)
−0.374607 0.927184i \(-0.622222\pi\)
\(264\) 0.223131 + 0.162114i 0.223131 + 0.162114i
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.517565 0.896448i −0.517565 0.896448i
\(269\) 0.876742 0.876742 0.438371 0.898794i \(-0.355556\pi\)
0.438371 + 0.898794i \(0.355556\pi\)
\(270\) −1.18529 + 3.64795i −1.18529 + 3.64795i
\(271\) 0.0697990 0.0697990 0.0348995 0.999391i \(-0.488889\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(272\) −3.75596 6.50552i −3.75596 6.50552i
\(273\) 0 0
\(274\) 0 0
\(275\) −0.0940924 + 0.162973i −0.0940924 + 0.162973i
\(276\) 0 0
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 0.270928 + 0.833831i 0.270928 + 0.833831i
\(280\) 0 0
\(281\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(282\) 0 0
\(283\) 0.997564 1.72783i 0.997564 1.72783i 0.438371 0.898794i \(-0.355556\pi\)
0.559193 0.829038i \(-0.311111\pi\)
\(284\) −0.921036 + 1.59528i −0.921036 + 1.59528i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.80182 5.54543i −1.80182 5.54543i
\(289\) 1.34730 1.34730
\(290\) −2.36148 4.09020i −2.36148 4.09020i
\(291\) 0 0
\(292\) 0 0
\(293\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(294\) −1.82264 + 0.811492i −1.82264 + 0.811492i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.0682737 + 0.0145120i −0.0682737 + 0.0145120i
\(298\) 0 0
\(299\) 0 0
\(300\) 7.34108 3.26846i 7.34108 3.26846i
\(301\) 0 0
\(302\) 0 0
\(303\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(304\) 0 0
\(305\) 2.94548 2.94548
\(306\) 2.98992 + 0.635526i 2.98992 + 0.635526i
\(307\) 1.92252 1.92252 0.961262 0.275637i \(-0.0888889\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.68145 2.91236i 1.68145 2.91236i
\(311\) 0.939693 1.62760i 0.939693 1.62760i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(312\) 0 0
\(313\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(314\) 2.45665 2.45665
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) 0 0
\(319\) 0.0429726 0.0744306i 0.0429726 0.0744306i
\(320\) −6.46944 + 11.2054i −6.46944 + 11.2054i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 2.72286 + 1.21229i 2.72286 + 1.21229i
\(325\) 0 0
\(326\) −1.43518 2.48580i −1.43518 2.48580i
\(327\) 1.22256 0.544320i 1.22256 0.544320i
\(328\) 0 0
\(329\) 0 0
\(330\) 0.216595 + 0.157366i 0.216595 + 0.157366i
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) −2.23306 −2.23306
\(333\) 0 0
\(334\) 0 0
\(335\) −0.333843 0.578232i −0.333843 0.578232i
\(336\) 0 0
\(337\) −0.961262 + 1.66495i −0.961262 + 1.66495i −0.241922 + 0.970296i \(0.577778\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(338\) 0.997564 1.72783i 0.997564 1.72783i
\(339\) 0.0218524 0.207912i 0.0218524 0.207912i
\(340\) −4.38955 7.60292i −4.38955 7.60292i
\(341\) 0.0611957 0.0611957
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.438371 + 0.759281i −0.438371 + 0.759281i −0.997564 0.0697565i \(-0.977778\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(348\) −3.35271 + 1.49272i −3.35271 + 1.49272i
\(349\) 0.882948 + 1.52931i 0.882948 + 1.52931i 0.848048 + 0.529919i \(0.177778\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.406985 −0.406985
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) −0.594092 + 1.02900i −0.594092 + 1.02900i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(360\) −2.34751 7.22489i −2.34751 7.22489i
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 0.104019 0.989677i 0.104019 0.989677i
\(364\) 0 0
\(365\) 0 0
\(366\) 0.319514 3.03997i 0.319514 3.03997i
\(367\) 0.374607 + 0.648838i 0.374607 + 0.648838i 0.990268 0.139173i \(-0.0444444\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.11409 1.53598i −2.11409 1.53598i
\(373\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(374\) 0.106678 0.184771i 0.106678 0.184771i
\(375\) 2.97887 1.32628i 2.97887 1.32628i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −1.61323 + 0.718254i −1.61323 + 0.718254i
\(382\) 0 0
\(383\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(384\) 6.14586 + 4.46523i 6.14586 + 4.46523i
\(385\) 0 0
\(386\) 2.87035 2.87035
\(387\) 0 0
\(388\) 0 0
\(389\) −0.559193 0.968551i −0.559193 0.968551i −0.997564 0.0697565i \(-0.977778\pi\)
0.438371 0.898794i \(-0.355556\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.97571 3.42203i 1.97571 3.42203i
\(393\) 0 0
\(394\) 1.69196 + 2.93057i 1.69196 + 2.93057i
\(395\) 0 0
\(396\) 0.139205 0.154603i 0.139205 0.154603i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −6.60956 + 11.4481i −6.60956 + 11.4481i
\(401\) 0.374607 0.648838i 0.374607 0.648838i −0.615661 0.788011i \(-0.711111\pi\)
0.990268 + 0.139173i \(0.0444444\pi\)
\(402\) −0.632996 + 0.281828i −0.632996 + 0.281828i
\(403\) 0 0
\(404\) −4.82261 −4.82261
\(405\) 1.75631 + 0.781961i 1.75631 + 0.781961i
\(406\) 0 0
\(407\) 0 0
\(408\) −5.53054 + 2.46236i −5.53054 + 2.46236i
\(409\) 0.241922 0.419021i 0.241922 0.419021i −0.719340 0.694658i \(-0.755556\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.44038 −1.44038
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.990268 + 1.71519i −0.990268 + 1.71519i −0.374607 + 0.927184i \(0.622222\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(420\) 0 0
\(421\) 0.104528 + 0.181049i 0.104528 + 0.181049i 0.913545 0.406737i \(-0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(422\) 1.49478 1.49478
\(423\) 0 0
\(424\) 0 0
\(425\) −2.06533 3.57726i −2.06533 3.57726i
\(426\) 0.997564 + 0.724773i 0.997564 + 0.724773i
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.43868 −1.43868 −0.719340 0.694658i \(-0.755556\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(432\) −4.79592 + 1.01940i −4.79592 + 1.01940i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −2.16259 + 0.962846i −2.16259 + 0.962846i
\(436\) −1.99437 + 3.45435i −1.99437 + 3.45435i
\(437\) 0 0
\(438\) 0 0
\(439\) −0.913545 1.58231i −0.913545 1.58231i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(440\) −0.530242 −0.530242
\(441\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(442\) 0 0
\(443\) 0.241922 + 0.419021i 0.241922 + 0.419021i 0.961262 0.275637i \(-0.0888889\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −1.66222 5.11579i −1.66222 5.11579i
\(451\) 0 0
\(452\) 0.311551 + 0.539622i 0.311551 + 0.539622i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(458\) 0 0
\(459\) 0.473442 1.45710i 0.473442 1.45710i
\(460\) 0 0
\(461\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(462\) 0 0
\(463\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 3.01862 5.22841i 3.01862 5.22841i
\(465\) −1.36364 0.990746i −1.36364 0.990746i
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.128708 1.22458i 0.128708 1.22458i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1.99513 −1.99513
\(479\) −0.559193 0.968551i −0.559193 0.968551i −0.997564 0.0697565i \(-0.977778\pi\)
0.438371 0.898794i \(-0.355556\pi\)
\(480\) 9.06897 + 6.58899i 9.06897 + 6.58899i
\(481\) 0 0
\(482\) −0.208548 + 0.361215i −0.208548 + 0.361215i
\(483\) 0 0
\(484\) 1.48301 + 2.56864i 1.48301 + 2.56864i
\(485\) 0 0
\(486\) 0.997564 1.72783i 0.997564 1.72783i
\(487\) −1.87939 −1.87939 −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(488\) 3.02697 + 5.24286i 3.02697 + 5.24286i
\(489\) −1.31430 + 0.585164i −1.31430 + 0.585164i
\(490\) 1.91784 3.32180i 1.91784 3.32180i
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0.943248 + 1.63375i 0.943248 + 1.63375i
\(494\) 0 0
\(495\) 0.0897908 0.0997228i 0.0897908 0.0997228i
\(496\) 4.29872 4.29872
\(497\) 0 0
\(498\) −0.156247 + 1.48659i −0.156247 + 1.48659i
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) −4.85944 + 8.41680i −4.85944 + 8.41680i
\(501\) 0 0
\(502\) −1.87481 3.24726i −1.87481 3.24726i
\(503\) 0.347296 0.347296 0.173648 0.984808i \(-0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(504\) 0 0
\(505\) −3.11071 −3.11071
\(506\) 0 0
\(507\) −0.809017 0.587785i −0.809017 0.587785i
\(508\) 2.63166 4.55816i 2.63166 4.55816i
\(509\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(510\) −5.36854 + 2.39023i −5.36854 + 2.39023i
\(511\) 0 0
\(512\) −9.21474 −9.21474
\(513\) 0 0
\(514\) 3.52319 3.52319
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0.759146 + 2.33641i 0.759146 + 2.33641i
\(523\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 1.97571 3.42203i 1.97571 3.42203i
\(527\) −0.671624 + 1.16329i −0.671624 + 1.16329i
\(528\) −0.0357726 + 0.340354i −0.0357726 + 0.340354i
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.686157 1.18846i 0.686157 1.18846i
\(537\) 0 0
\(538\) 0.874607 + 1.51486i 0.874607 + 1.51486i
\(539\) 0.0697990 0.0697990
\(540\) −5.60493 + 1.19137i −5.60493 + 1.19137i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0.0696290 + 0.120601i 0.0696290 + 0.120601i
\(543\) 0 0
\(544\) 4.46666 7.73648i 4.46666 7.73648i
\(545\) −1.28642 + 2.22814i −1.28642 + 2.22814i
\(546\) 0 0
\(547\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(548\) 0 0
\(549\) −1.49861 0.318539i −1.49861 0.318539i
\(550\) −0.375453 −0.375453
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −1.17045 + 1.29992i −1.17045 + 1.29992i
\(559\) 0 0
\(560\) 0 0
\(561\) −0.0865149 0.0628567i −0.0865149 0.0628567i
\(562\) 0 0
\(563\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0.200958 + 0.348070i 0.200958 + 0.348070i
\(566\) 3.98054 3.98054
\(567\) 0 0
\(568\) −2.44211 −2.44211
\(569\) −0.438371 0.759281i −0.438371 0.759281i 0.559193 0.829038i \(-0.311111\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(570\) 0 0
\(571\) 0.978148 1.69420i 0.978148 1.69420i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 4.50335 5.00148i 4.50335 5.00148i
\(577\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(578\) 1.34401 + 2.32790i 1.34401 + 2.32790i
\(579\) 0.150383 1.43080i 0.150383 1.43080i
\(580\) 3.52783 6.11039i 3.52783 6.11039i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −1.23306 −1.23306
\(587\) 0.719340 + 1.24593i 0.719340 + 1.24593i 0.961262 + 0.275637i \(0.0888889\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(588\) −2.41130 1.75192i −2.41130 1.75192i
\(589\) 0 0
\(590\) 0 0
\(591\) 1.54946 0.689864i 1.54946 0.689864i
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) −0.0931817 0.103489i −0.0931817 0.103489i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −0.913545 + 1.58231i −0.913545 + 1.58231i −0.104528 + 0.994522i \(0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(600\) 8.61880 + 6.26192i 8.61880 + 6.26192i
\(601\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 0 0
\(603\) 0.107320 + 0.330298i 0.107320 + 0.330298i
\(604\) 0 0
\(605\) 0.956579 + 1.65684i 0.956579 + 1.65684i
\(606\) −0.337437 + 3.21050i −0.337437 + 3.21050i
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 2.93830 + 5.08929i 2.93830 + 5.08929i
\(611\) 0 0
\(612\) 1.41111 + 4.34295i 1.41111 + 4.34295i
\(613\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 1.91784 + 3.32180i 1.91784 + 3.32180i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 5.02387 5.02387
\(621\) 0 0
\(622\) 3.74961 3.74961
\(623\) 0 0
\(624\) 0 0
\(625\) −1.78642 + 3.09417i −1.78642 + 3.09417i
\(626\) 0 0
\(627\) 0 0
\(628\) 1.83500 + 3.17832i 1.83500 + 3.17832i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.92252 1.92252 0.961262 0.275637i \(-0.0888889\pi\)
0.961262 + 0.275637i \(0.0888889\pi\)
\(632\) 0 0
\(633\) 0.0783141 0.745109i 0.0783141 0.745109i
\(634\) 0 0
\(635\) 1.69749 2.94013i 1.69749 2.94013i
\(636\) 0 0
\(637\) 0 0
\(638\) 0.171471 0.171471
\(639\) 0.413545 0.459289i 0.413545 0.459289i
\(640\) −14.6048 −14.6048
\(641\) −0.848048 1.46886i −0.848048 1.46886i −0.882948 0.469472i \(-0.844444\pi\)
0.0348995 0.999391i \(-0.488889\pi\)
\(642\) 0 0
\(643\) 0.882948 1.52931i 0.882948 1.52931i 0.0348995 0.999391i \(-0.488889\pi\)
0.848048 0.529919i \(-0.177778\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.11839 1.11839 0.559193 0.829038i \(-0.311111\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(648\) 0.413036 + 3.92978i 0.413036 + 3.92978i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 2.14402 3.71355i 2.14402 3.71355i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 2.16008 + 1.56939i 2.16008 + 1.56939i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) −0.0418071 + 0.397768i −0.0418071 + 0.397768i
\(661\) −0.0348995 + 0.0604477i −0.0348995 + 0.0604477i −0.882948 0.469472i \(-0.844444\pi\)
0.848048 + 0.529919i \(0.177778\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −1.48023 2.56383i −1.48023 2.56383i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0.666059 1.15365i 0.666059 1.15365i
\(671\) −0.0534691 + 0.0926113i −0.0534691 + 0.0926113i
\(672\) 0 0
\(673\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) −3.83568 −3.83568
\(675\) −2.63718 + 0.560550i −2.63718 + 0.560550i
\(676\) 2.98054 2.98054
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0.381036 0.169648i 0.381036 0.169648i
\(679\) 0 0
\(680\) 5.81941 10.0795i 5.81941 10.0795i
\(681\) 0 0
\(682\) 0.0610467 + 0.105736i 0.0610467 + 0.105736i
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.939693 + 1.62760i 0.939693 + 1.62760i 0.766044 + 0.642788i \(0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.74921 −1.74921
\(695\) 0 0
\(696\) −3.93625 2.85986i −3.93625 2.85986i
\(697\) 0 0
\(698\) −1.76159 + 3.05117i −1.76159 + 3.05117i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.234879 0.406822i −0.234879 0.406822i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) −2.37058 −2.37058
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(718\) −1.61409 2.79569i −1.61409 2.79569i
\(719\) −0.483844 −0.483844 −0.241922 0.970296i \(-0.577778\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(720\) 6.30739 7.00507i 6.30739 7.00507i
\(721\) 0 0
\(722\) 0.997564 + 1.72783i 0.997564 + 1.72783i
\(723\) 0.169131 + 0.122881i 0.169131 + 0.122881i
\(724\) 0 0
\(725\) 1.65988 2.87500i 1.65988 2.87500i
\(726\) 1.81376 0.807538i 1.81376 0.807538i
\(727\) −0.913545 1.58231i −0.913545 1.58231i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(728\) 0 0
\(729\) −0.809017 0.587785i −0.809017 0.587785i
\(730\) 0 0
\(731\) 0 0
\(732\) 4.17166 1.85734i 4.17166 1.85734i
\(733\) −0.438371 + 0.759281i −0.438371 + 0.759281i −0.997564 0.0697565i \(-0.977778\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(734\) −0.747388 + 1.29451i −0.747388 + 1.29451i
\(735\) −1.55535 1.13003i −1.55535 1.13003i
\(736\) 0 0
\(737\) 0.0242409 0.0242409
\(738\) 0 0
\(739\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 0.362126 3.44540i 0.362126 3.44540i
\(745\) 0 0
\(746\) −1.23306 −1.23306
\(747\) 0.732841 + 0.155770i 0.732841 + 0.155770i
\(748\) 0.318733 0.318733
\(749\) 0 0
\(750\) 5.26321 + 3.82394i 5.26321 + 3.82394i
\(751\) −0.669131 + 1.15897i −0.669131 + 1.15897i 0.309017 + 0.951057i \(0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(752\) 0 0
\(753\) −1.71690 + 0.764415i −1.71690 + 0.764415i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.99513 −1.99513 −0.997564 0.0697565i \(-0.977778\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.669131 + 1.15897i −0.669131 + 1.15897i 0.309017 + 0.951057i \(0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(762\) −2.85032 2.07088i −2.85032 2.07088i
\(763\) 0 0
\(764\) 0 0
\(765\) 0.910202 + 2.80131i 0.910202 + 2.80131i
\(766\) 1.99513 1.99513
\(767\) 0 0
\(768\) −0.880783 + 8.38009i −0.880783 + 8.38009i
\(769\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(770\) 0 0
\(771\) 0.184586 1.75622i 0.184586 1.75622i
\(772\) 2.14402 + 3.71355i 2.14402 + 3.71355i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 2.36378 2.36378
\(776\) 0 0
\(777\) 0 0
\(778\) 1.11566 1.93238i 1.11566 1.93238i
\(779\) 0 0
\(780\) 0 0
\(781\) −0.0215691 0.0373587i −0.0215691 0.0373587i
\(782\) 0 0
\(783\) 1.20442 0.256006i 1.20442 0.256006i
\(784\) 4.90306 4.90306
\(785\) 1.18362 + 2.05010i 1.18362 + 2.05010i
\(786\) 0 0
\(787\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(788\) −2.52764 + 4.37800i −2.52764 + 4.37800i
\(789\) −1.60229 1.16413i −1.60229 1.16413i
\(790\) 0 0
\(791\) 0 0
\(792\) 0.269778 + 0.0573432i 0.269778 + 0.0573432i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.559193 + 0.968551i −0.559193 + 0.968551i 0.438371 + 0.898794i \(0.355556\pi\)
−0.997564 + 0.0697565i \(0.977778\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −15.7204 −15.7204
\(801\) 0 0
\(802\) 1.49478 1.49478
\(803\) 0 0
\(804\) −0.837437 0.608434i −0.837437 0.608434i
\(805\) 0 0
\(806\) 0 0
\(807\) 0.800944 0.356603i 0.800944 0.356603i
\(808\) −3.19677 5.53697i −3.19677 5.53697i
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0.400938 + 3.81467i 0.400938 + 3.81467i
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0.0637646 0.0283898i 0.0637646 0.0283898i
\(814\) 0 0
\(815\) 1.38295 2.39534i 1.38295 2.39534i
\(816\) −6.07727 4.41540i −6.07727 4.41540i
\(817\) 0 0
\(818\) 0.965330 0.965330
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 0 0
\(825\) −0.0196707 + 0.187154i −0.0196707 + 0.187154i
\(826\) 0 0
\(827\) −1.23132 −1.23132 −0.615661 0.788011i \(-0.711111\pi\)
−0.615661 + 0.788011i \(0.711111\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) −1.43687 2.48873i −1.43687 2.48873i
\(831\) 0 0
\(832\) 0 0
\(833\) −0.766044 + 1.32683i −0.766044 + 1.32683i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.586655 + 0.651546i 0.586655 + 0.651546i
\(838\) −3.95142 −3.95142
\(839\) −0.848048 1.46886i −0.848048 1.46886i −0.882948 0.469472i \(-0.844444\pi\)
0.0348995 0.999391i \(-0.488889\pi\)
\(840\) 0 0
\(841\) −0.258078 + 0.447004i −0.258078 + 0.447004i
\(842\) −0.208548 + 0.361215i −0.208548 + 0.361215i
\(843\) 0 0
\(844\) 1.11653 + 1.93388i 1.11653 + 1.93388i
\(845\) 1.92252 1.92252
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.208548 1.98420i 0.208548 1.98420i
\(850\) 4.12060 7.13708i 4.12060 7.13708i
\(851\) 0 0
\(852\) −0.192549 + 1.83198i −0.192549 + 1.83198i
\(853\) 0.241922 + 0.419021i 0.241922 + 0.419021i 0.961262 0.275637i \(-0.0888889\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) 0 0
\(859\) −0.848048 + 1.46886i −0.848048 + 1.46886i 0.0348995 + 0.999391i \(0.488889\pi\)
−0.882948 + 0.469472i \(0.844444\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.43518 2.48580i −1.43518 2.48580i
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −3.90157 4.33314i −3.90157 4.33314i
\(865\) 0 0
\(866\) 0 0
\(867\) 1.23082 0.547995i 1.23082 0.547995i
\(868\) 0 0
\(869\) 0 0
\(870\) −3.82096 2.77609i −3.82096 2.77609i
\(871\) 0 0
\(872\) −5.28804 −5.28804
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.374607 0.648838i 0.374607 0.648838i −0.615661 0.788011i \(-0.711111\pi\)
0.990268 + 0.139173i \(0.0444444\pi\)
\(878\) 1.82264 3.15691i 1.82264 3.15691i
\(879\) −0.0646021 + 0.614648i −0.0646021 + 0.614648i
\(880\) −0.328971 0.569795i −0.328971 0.569795i
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −1.33500 + 1.48267i −1.33500 + 1.48267i
\(883\) 1.11839 1.11839 0.559193 0.829038i \(-0.311111\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.482665 + 0.836001i −0.482665 + 0.836001i
\(887\) 0.978148 1.69420i 0.978148 1.69420i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.0564686 + 0.0410268i −0.0564686 + 0.0410268i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.07955 −1.07955
\(900\) 5.37701 5.97177i 5.37701 5.97177i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −0.413036 + 0.715400i −0.413036 + 0.715400i
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(908\) 0 0
\(909\) 1.58268 + 0.336408i 1.58268 + 0.336408i
\(910\) 0 0
\(911\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(912\) 0 0
\(913\) 0.0261472 0.0452882i 0.0261472 0.0452882i
\(914\) 0.616528 1.06786i 0.616528 1.06786i
\(915\) 2.69083 1.19803i 2.69083 1.19803i
\(916\) 0 0
\(917\) 0 0
\(918\) 2.98992 0.635526i 2.98992 0.635526i
\(919\) −1.43868 −1.43868 −0.719340 0.694658i \(-0.755556\pi\)
−0.719340 + 0.694658i \(0.755556\pi\)
\(920\) 0 0
\(921\) 1.75631 0.781961i 1.75631 0.781961i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 7.17961 7.17961
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0.351519 3.34448i 0.351519 3.34448i
\(931\) 0 0
\(932\) 0 0
\(933\) 0.196449 1.86909i 0.196449 1.86909i
\(934\) 0 0
\(935\) 0.205591 0.205591
\(936\) 0 0
\(937\) 1.11839 1.11839 0.559193 0.829038i \(-0.311111\pi\)
0.559193 + 0.829038i \(0.311111\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 2.24426 0.999208i 2.24426 0.999208i
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.49027 2.58122i −1.49027 2.58122i
\(957\) 0.00898371 0.0854743i 0.00898371 0.0854743i
\(958\) 1.11566 1.93238i 1.11566 1.93238i
\(959\) 0 0
\(960\) −1.35248 + 12.8680i −1.35248 + 12.8680i
\(961\) 0.115661 + 0.200332i 0.115661 + 0.200332i
\(962\) 0 0
\(963\) 0 0
\(964\) −0.623102 −0.623102
\(965\) 1.38295 + 2.39534i 1.38295 + 2.39534i
\(966\) 0 0
\(967\) 0.978148 1.69420i 0.978148 1.69420i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(968\) −1.96609 + 3.40536i −1.96609 + 3.40536i
\(969\) 0 0
\(970\) 0 0
\(971\) −0.483844 −0.483844 −0.241922 0.970296i \(-0.577778\pi\)
−0.241922 + 0.970296i \(0.577778\pi\)
\(972\) 2.98054 2.98054
\(973\) 0 0
\(974\) −1.87481 3.24726i −1.87481 3.24726i
\(975\) 0 0
\(976\) −3.75596 + 6.50552i −3.75596 + 6.50552i
\(977\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) −2.32216 1.68715i −2.32216 1.68715i
\(979\) 0 0
\(980\) 5.73015 5.73015
\(981\) 0.895472 0.994522i 0.895472 0.994522i
\(982\) 0 0
\(983\) 0.882948 + 1.52931i 0.882948 + 1.52931i 0.848048 + 0.529919i \(0.177778\pi\)
0.0348995 + 0.999391i \(0.488889\pi\)
\(984\) 0 0
\(985\) −1.63039 + 2.82392i −1.63039 + 2.82392i
\(986\) −1.88190 + 3.25955i −1.88190 + 3.25955i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0.261876 + 0.0556635i 0.261876 + 0.0556635i
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 2.55606 + 4.42722i 2.55606 + 4.42722i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) −2.04000 + 0.908266i −2.04000 + 0.908266i
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) 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 2151.1.f.b.238.12 24
9.7 even 3 inner 2151.1.f.b.1672.12 yes 24
239.238 odd 2 CM 2151.1.f.b.238.12 24
2151.1672 odd 6 inner 2151.1.f.b.1672.12 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.1.f.b.238.12 24 1.1 even 1 trivial
2151.1.f.b.238.12 24 239.238 odd 2 CM
2151.1.f.b.1672.12 yes 24 9.7 even 3 inner
2151.1.f.b.1672.12 yes 24 2151.1672 odd 6 inner