Properties

Label 1520.1.bh.c.189.8
Level $1520$
Weight $1$
Character 1520.189
Analytic conductor $0.759$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -95
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,1,Mod(189,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.189");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1520.bh (of order \(4\), 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: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

Embedding invariants

Embedding label 189.8
Root \(0.555570 - 0.831470i\) of defining polynomial
Character \(\chi\) \(=\) 1520.189
Dual form 1520.1.bh.c.949.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.980785 + 0.195090i) q^{2} +(-0.275899 - 0.275899i) q^{3} +(0.923880 + 0.382683i) q^{4} +(0.707107 - 0.707107i) q^{5} +(-0.216773 - 0.324423i) q^{6} +(0.831470 + 0.555570i) q^{8} -0.847759i q^{9} +O(q^{10})\) \(q+(0.980785 + 0.195090i) q^{2} +(-0.275899 - 0.275899i) q^{3} +(0.923880 + 0.382683i) q^{4} +(0.707107 - 0.707107i) q^{5} +(-0.216773 - 0.324423i) q^{6} +(0.831470 + 0.555570i) q^{8} -0.847759i q^{9} +(0.831470 - 0.555570i) q^{10} +(-0.541196 + 0.541196i) q^{11} +(-0.149316 - 0.360480i) q^{12} +(-0.785695 - 0.785695i) q^{13} -0.390181 q^{15} +(0.707107 + 0.707107i) q^{16} +(0.165390 - 0.831470i) q^{18} +(0.707107 + 0.707107i) q^{19} +(0.923880 - 0.382683i) q^{20} +(-0.636379 + 0.425215i) q^{22} +(-0.0761205 - 0.382683i) q^{24} -1.00000i q^{25} +(-0.617317 - 0.923880i) q^{26} +(-0.509796 + 0.509796i) q^{27} +(-0.382683 - 0.0761205i) q^{30} +(0.555570 + 0.831470i) q^{32} +0.298631 q^{33} +(0.324423 - 0.783227i) q^{36} +(-1.38704 + 1.38704i) q^{37} +(0.555570 + 0.831470i) q^{38} +0.433546i q^{39} +(0.980785 - 0.195090i) q^{40} +(-0.707107 + 0.292893i) q^{44} +(-0.599456 - 0.599456i) q^{45} -0.390181i q^{48} -1.00000 q^{49} +(0.195090 - 0.980785i) q^{50} +(-0.425215 - 1.02656i) q^{52} +(0.275899 - 0.275899i) q^{53} +(-0.599456 + 0.400544i) q^{54} +0.765367i q^{55} -0.390181i q^{57} +(-0.360480 - 0.149316i) q^{60} +(0.541196 + 0.541196i) q^{61} +(0.382683 + 0.923880i) q^{64} -1.11114 q^{65} +(0.292893 + 0.0582601i) q^{66} +(1.17588 + 1.17588i) q^{67} +(0.470990 - 0.704886i) q^{72} +(-1.63099 + 1.08979i) q^{74} +(-0.275899 + 0.275899i) q^{75} +(0.382683 + 0.923880i) q^{76} +(-0.0845805 + 0.425215i) q^{78} +1.00000 q^{80} -0.566454 q^{81} +(-0.750661 + 0.149316i) q^{88} +(-0.470990 - 0.704886i) q^{90} +1.00000 q^{95} +(0.0761205 - 0.382683i) q^{96} +1.11114 q^{97} +(-0.980785 - 0.195090i) q^{98} +(0.458804 + 0.458804i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{24} - 16 q^{26} - 16 q^{49} + 16 q^{66} + 16 q^{80} - 16 q^{81} + 16 q^{95} + 16 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(3\) −0.275899 0.275899i −0.275899 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(4\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(5\) 0.707107 0.707107i 0.707107 0.707107i
\(6\) −0.216773 0.324423i −0.216773 0.324423i
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(9\) 0.847759i 0.847759i
\(10\) 0.831470 0.555570i 0.831470 0.555570i
\(11\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(12\) −0.149316 0.360480i −0.149316 0.360480i
\(13\) −0.785695 0.785695i −0.785695 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(14\) 0 0
\(15\) −0.390181 −0.390181
\(16\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0.165390 0.831470i 0.165390 0.831470i
\(19\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(20\) 0.923880 0.382683i 0.923880 0.382683i
\(21\) 0 0
\(22\) −0.636379 + 0.425215i −0.636379 + 0.425215i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −0.0761205 0.382683i −0.0761205 0.382683i
\(25\) 1.00000i 1.00000i
\(26\) −0.617317 0.923880i −0.617317 0.923880i
\(27\) −0.509796 + 0.509796i −0.509796 + 0.509796i
\(28\) 0 0
\(29\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(30\) −0.382683 0.0761205i −0.382683 0.0761205i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(33\) 0.298631 0.298631
\(34\) 0 0
\(35\) 0 0
\(36\) 0.324423 0.783227i 0.324423 0.783227i
\(37\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(38\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(39\) 0.433546i 0.433546i
\(40\) 0.980785 0.195090i 0.980785 0.195090i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(45\) −0.599456 0.599456i −0.599456 0.599456i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0.390181i 0.390181i
\(49\) −1.00000 −1.00000
\(50\) 0.195090 0.980785i 0.195090 0.980785i
\(51\) 0 0
\(52\) −0.425215 1.02656i −0.425215 1.02656i
\(53\) 0.275899 0.275899i 0.275899 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(54\) −0.599456 + 0.400544i −0.599456 + 0.400544i
\(55\) 0.765367i 0.765367i
\(56\) 0 0
\(57\) 0.390181i 0.390181i
\(58\) 0 0
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) −0.360480 0.149316i −0.360480 0.149316i
\(61\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(65\) −1.11114 −1.11114
\(66\) 0.292893 + 0.0582601i 0.292893 + 0.0582601i
\(67\) 1.17588 + 1.17588i 1.17588 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0.470990 0.704886i 0.470990 0.704886i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −1.63099 + 1.08979i −1.63099 + 1.08979i
\(75\) −0.275899 + 0.275899i −0.275899 + 0.275899i
\(76\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(77\) 0 0
\(78\) −0.0845805 + 0.425215i −0.0845805 + 0.425215i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.00000 1.00000
\(81\) −0.566454 −0.566454
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.750661 + 0.149316i −0.750661 + 0.149316i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −0.470990 0.704886i −0.470990 0.704886i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 1.00000
\(96\) 0.0761205 0.382683i 0.0761205 0.382683i
\(97\) 1.11114 1.11114 0.555570 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(98\) −0.980785 0.195090i −0.980785 0.195090i
\(99\) 0.458804 + 0.458804i 0.458804 + 0.458804i
\(100\) 0.382683 0.923880i 0.382683 0.923880i
\(101\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(102\) 0 0
\(103\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(104\) −0.216773 1.08979i −0.216773 1.08979i
\(105\) 0 0
\(106\) 0.324423 0.216773i 0.324423 0.216773i
\(107\) −1.17588 + 1.17588i −1.17588 + 1.17588i −0.195090 + 0.980785i \(0.562500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(108\) −0.666080 + 0.275899i −0.666080 + 0.275899i
\(109\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(110\) −0.149316 + 0.750661i −0.149316 + 0.750661i
\(111\) 0.765367 0.765367
\(112\) 0 0
\(113\) 0.390181 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(114\) 0.0761205 0.382683i 0.0761205 0.382683i
\(115\) 0 0
\(116\) 0 0
\(117\) −0.666080 + 0.666080i −0.666080 + 0.666080i
\(118\) 0 0
\(119\) 0 0
\(120\) −0.324423 0.216773i −0.324423 0.216773i
\(121\) 0.414214i 0.414214i
\(122\) 0.425215 + 0.636379i 0.425215 + 0.636379i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.707107 0.707107i −0.707107 0.707107i
\(126\) 0 0
\(127\) −1.66294 −1.66294 −0.831470 0.555570i \(-0.812500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(128\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(129\) 0 0
\(130\) −1.08979 0.216773i −1.08979 0.216773i
\(131\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(132\) 0.275899 + 0.114281i 0.275899 + 0.114281i
\(133\) 0 0
\(134\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(135\) 0.720960i 0.720960i
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.850430 0.850430
\(144\) 0.599456 0.599456i 0.599456 0.599456i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.275899 + 0.275899i 0.275899 + 0.275899i
\(148\) −1.81225 + 0.750661i −1.81225 + 0.750661i
\(149\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(150\) −0.324423 + 0.216773i −0.324423 + 0.216773i
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −0.165911 + 0.400544i −0.165911 + 0.400544i
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) −0.152241 −0.152241
\(160\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(161\) 0 0
\(162\) −0.555570 0.110510i −0.555570 0.110510i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0.211164 0.211164i 0.211164 0.211164i
\(166\) 0 0
\(167\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(168\) 0 0
\(169\) 0.234633i 0.234633i
\(170\) 0 0
\(171\) 0.599456 0.599456i 0.599456 0.599456i
\(172\) 0 0
\(173\) −1.17588 1.17588i −1.17588 1.17588i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.765367 −0.765367
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(180\) −0.324423 0.783227i −0.324423 0.783227i
\(181\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(182\) 0 0
\(183\) 0.298631i 0.298631i
\(184\) 0 0
\(185\) 1.96157i 1.96157i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(191\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(192\) 0.149316 0.360480i 0.149316 0.360480i
\(193\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(194\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(195\) 0.306563 + 0.306563i 0.306563 + 0.306563i
\(196\) −0.923880 0.382683i −0.923880 0.382683i
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0.360480 + 0.539496i 0.360480 + 0.539496i
\(199\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) 0.555570 0.831470i 0.555570 0.831470i
\(201\) 0.648847i 0.648847i
\(202\) 0.636379 0.425215i 0.636379 0.425215i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0.382683 1.92388i 0.382683 1.92388i
\(207\) 0 0
\(208\) 1.11114i 1.11114i
\(209\) −0.765367 −0.765367
\(210\) 0 0
\(211\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(212\) 0.360480 0.149316i 0.360480 0.149316i
\(213\) 0 0
\(214\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(215\) 0 0
\(216\) −0.707107 + 0.140652i −0.707107 + 0.140652i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(221\) 0 0
\(222\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(223\) 1.11114 1.11114 0.555570 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(224\) 0 0
\(225\) −0.847759 −0.847759
\(226\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(227\) −0.785695 0.785695i −0.785695 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(228\) 0.149316 0.360480i 0.149316 0.360480i
\(229\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −0.783227 + 0.523336i −0.783227 + 0.523336i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −0.275899 0.275899i −0.275899 0.275899i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −0.0808091 + 0.406255i −0.0808091 + 0.406255i
\(243\) 0.666080 + 0.666080i 0.666080 + 0.666080i
\(244\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(245\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(246\) 0 0
\(247\) 1.11114i 1.11114i
\(248\) 0 0
\(249\) 0 0
\(250\) −0.555570 0.831470i −0.555570 0.831470i
\(251\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.63099 0.324423i −1.63099 0.324423i
\(255\) 0 0
\(256\) 1.00000i 1.00000i
\(257\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.02656 0.425215i −1.02656 0.425215i
\(261\) 0 0
\(262\) −0.785695 1.17588i −0.785695 1.17588i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0.248303 + 0.165911i 0.248303 + 0.165911i
\(265\) 0.390181i 0.390181i
\(266\) 0 0
\(267\) 0 0
\(268\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(269\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(270\) −0.140652 + 0.707107i −0.140652 + 0.707107i
\(271\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) −1.53636 + 1.02656i −1.53636 + 1.02656i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) −0.275899 0.275899i −0.275899 0.275899i
\(286\) 0.834089 + 0.165911i 0.834089 + 0.165911i
\(287\) 0 0
\(288\) 0.704886 0.470990i 0.704886 0.470990i
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) −0.306563 0.306563i −0.306563 0.306563i
\(292\) 0 0
\(293\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(294\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(295\) 0 0
\(296\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(297\) 0.551799i 0.551799i
\(298\) −1.53636 + 1.02656i −1.53636 + 1.02656i
\(299\) 0 0
\(300\) −0.360480 + 0.149316i −0.360480 + 0.149316i
\(301\) 0 0
\(302\) 0 0
\(303\) −0.298631 −0.298631
\(304\) 1.00000i 1.00000i
\(305\) 0.765367 0.765367
\(306\) 0 0
\(307\) 0.785695 + 0.785695i 0.785695 + 0.785695i 0.980785 0.195090i \(-0.0625000\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(308\) 0 0
\(309\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(310\) 0 0
\(311\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(312\) −0.240865 + 0.360480i −0.240865 + 0.360480i
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.17588 1.17588i −1.17588 1.17588i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(318\) −0.149316 0.0297007i −0.149316 0.0297007i
\(319\) 0 0
\(320\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(321\) 0.648847 0.648847
\(322\) 0 0
\(323\) 0 0
\(324\) −0.523336 0.216773i −0.523336 0.216773i
\(325\) −0.785695 + 0.785695i −0.785695 + 0.785695i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0.248303 0.165911i 0.248303 0.165911i
\(331\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) 0 0
\(333\) 1.17588 + 1.17588i 1.17588 + 1.17588i
\(334\) 0.324423 1.63099i 0.324423 1.63099i
\(335\) 1.66294 1.66294
\(336\) 0 0
\(337\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(338\) −0.0457747 + 0.230125i −0.0457747 + 0.230125i
\(339\) −0.107651 0.107651i −0.107651 0.107651i
\(340\) 0 0
\(341\) 0 0
\(342\) 0.704886 0.470990i 0.704886 0.470990i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.923880 1.38268i −0.923880 1.38268i
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(350\) 0 0
\(351\) 0.801088 0.801088
\(352\) −0.750661 0.149316i −0.750661 0.149316i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(360\) −0.165390 0.831470i −0.165390 0.831470i
\(361\) 1.00000i 1.00000i
\(362\) 0 0
\(363\) 0.114281 0.114281i 0.114281 0.114281i
\(364\) 0 0
\(365\) 0 0
\(366\) 0.0582601 0.292893i 0.0582601 0.292893i
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(371\) 0 0
\(372\) 0 0
\(373\) −0.275899 + 0.275899i −0.275899 + 0.275899i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(374\) 0 0
\(375\) 0.390181i 0.390181i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(380\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(381\) 0.458804 + 0.458804i 0.458804 + 0.458804i
\(382\) −1.38704 0.275899i −1.38704 0.275899i
\(383\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(384\) 0.216773 0.324423i 0.216773 0.324423i
\(385\) 0 0
\(386\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(387\) 0 0
\(388\) 1.02656 + 0.425215i 1.02656 + 0.425215i
\(389\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(390\) 0.240865 + 0.360480i 0.240865 + 0.360480i
\(391\) 0 0
\(392\) −0.831470 0.555570i −0.831470 0.555570i
\(393\) 0.551799i 0.551799i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.248303 + 0.599456i 0.248303 + 0.599456i
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) −0.275899 + 1.38704i −0.275899 + 1.38704i
\(399\) 0 0
\(400\) 0.707107 0.707107i 0.707107 0.707107i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0.126584 0.636379i 0.126584 0.636379i
\(403\) 0 0
\(404\) 0.707107 0.292893i 0.707107 0.292893i
\(405\) −0.400544 + 0.400544i −0.400544 + 0.400544i
\(406\) 0 0
\(407\) 1.50132i 1.50132i
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.750661 1.81225i 0.750661 1.81225i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.216773 1.08979i 0.216773 1.08979i
\(417\) 0.720960 0.720960
\(418\) −0.750661 0.149316i −0.750661 0.149316i
\(419\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0.382683 0.0761205i 0.382683 0.0761205i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.53636 + 0.636379i −1.53636 + 0.636379i
\(429\) −0.234633 0.234633i −0.234633 0.234633i
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −0.720960 −0.720960
\(433\) −0.390181 −0.390181 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −0.425215 + 0.636379i −0.425215 + 0.636379i
\(441\) 0.847759i 0.847759i
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(445\) 0 0
\(446\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(447\) 0.720960 0.720960
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −0.831470 0.165390i −0.831470 0.165390i
\(451\) 0 0
\(452\) 0.360480 + 0.149316i 0.360480 + 0.149316i
\(453\) 0 0
\(454\) −0.617317 0.923880i −0.617317 0.923880i
\(455\) 0 0
\(456\) 0.216773 0.324423i 0.216773 0.324423i
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 1.53636 1.02656i 1.53636 1.02656i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) −0.870275 + 0.360480i −0.870275 + 0.360480i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.707107 0.707107i 0.707107 0.707107i
\(476\) 0 0
\(477\) −0.233896 0.233896i −0.233896 0.233896i
\(478\) 0 0
\(479\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(480\) −0.216773 0.324423i −0.216773 0.324423i
\(481\) 2.17958 2.17958
\(482\) 0 0
\(483\) 0 0
\(484\) −0.158513 + 0.382683i −0.158513 + 0.382683i
\(485\) 0.785695 0.785695i 0.785695 0.785695i
\(486\) 0.523336 + 0.783227i 0.523336 + 0.783227i
\(487\) 0.390181i 0.390181i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(488\) 0.149316 + 0.750661i 0.149316 + 0.750661i
\(489\) 0 0
\(490\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(491\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0.216773 1.08979i 0.216773 1.08979i
\(495\) 0.648847 0.648847
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(500\) −0.382683 0.923880i −0.382683 0.923880i
\(501\) −0.458804 + 0.458804i −0.458804 + 0.458804i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0.765367i 0.765367i
\(506\) 0 0
\(507\) 0.0647351 0.0647351i 0.0647351 0.0647351i
\(508\) −1.53636 0.636379i −1.53636 0.636379i
\(509\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(513\) −0.720960 −0.720960
\(514\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(515\) −1.38704 1.38704i −1.38704 1.38704i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0.648847i 0.648847i
\(520\) −0.923880 0.617317i −0.923880 0.617317i
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(524\) −0.541196 1.30656i −0.541196 1.30656i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.211164 + 0.211164i 0.211164 + 0.211164i
\(529\) −1.00000 −1.00000
\(530\) 0.0761205 0.382683i 0.0761205 0.382683i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.66294i 1.66294i
\(536\) 0.324423 + 1.63099i 0.324423 + 1.63099i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.541196 0.541196i 0.541196 0.541196i
\(540\) −0.275899 + 0.666080i −0.275899 + 0.666080i
\(541\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(542\) −1.81225 0.360480i −1.81225 0.360480i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(548\) 0 0
\(549\) 0.458804 0.458804i 0.458804 0.458804i
\(550\) 0.425215 + 0.636379i 0.425215 + 0.636379i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.541196 0.541196i 0.541196 0.541196i
\(556\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.38704 1.38704i −1.38704 1.38704i −0.831470 0.555570i \(-0.812500\pi\)
−0.555570 0.831470i \(-0.687500\pi\)
\(564\) 0 0
\(565\) 0.275899 0.275899i 0.275899 0.275899i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) −0.216773 0.324423i −0.216773 0.324423i
\(571\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(572\) 0.785695 + 0.325446i 0.785695 + 0.325446i
\(573\) 0.390181 + 0.390181i 0.390181 + 0.390181i
\(574\) 0 0
\(575\) 0 0
\(576\) 0.783227 0.324423i 0.783227 0.324423i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(579\) −0.458804 0.458804i −0.458804 0.458804i
\(580\) 0 0
\(581\) 0 0
\(582\) −0.240865 0.360480i −0.240865 0.360480i
\(583\) 0.298631i 0.298631i
\(584\) 0 0
\(585\) 0.941979i 0.941979i
\(586\) 1.63099 1.08979i 1.63099 1.08979i
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0.149316 + 0.360480i 0.149316 + 0.360480i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.96157 −1.96157
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0.107651 0.541196i 0.107651 0.541196i
\(595\) 0 0
\(596\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(597\) 0.390181 0.390181i 0.390181 0.390181i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0.996859 0.996859i 0.996859 0.996859i
\(604\) 0 0
\(605\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(606\) −0.292893 0.0582601i −0.292893 0.0582601i
\(607\) −0.390181 −0.390181 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(608\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(609\) 0 0
\(610\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) −0.636379 + 0.425215i −0.636379 + 0.425215i
\(619\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.149316 0.750661i 0.149316 0.750661i
\(623\) 0 0
\(624\) −0.306563 + 0.306563i −0.306563 + 0.306563i
\(625\) −1.00000 −1.00000
\(626\) 0 0
\(627\) 0.211164 + 0.211164i 0.211164 + 0.211164i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.923880 1.38268i −0.923880 1.38268i
\(635\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(636\) −0.140652 0.0582601i −0.140652 0.0582601i
\(637\) 0.785695 + 0.785695i 0.785695 + 0.785695i
\(638\) 0 0
\(639\) 0 0
\(640\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0.636379 + 0.126584i 0.636379 + 0.126584i
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −0.470990 0.314705i −0.470990 0.314705i
\(649\) 0 0
\(650\) −0.923880 + 0.617317i −0.923880 + 0.617317i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) −1.41421 −1.41421
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(660\) 0.275899 0.114281i 0.275899 0.114281i
\(661\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(667\) 0 0
\(668\) 0.636379 1.53636i 0.636379 1.53636i
\(669\) −0.306563 0.306563i −0.306563 0.306563i
\(670\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(671\) −0.585786 −0.585786
\(672\) 0 0
\(673\) −1.96157 −1.96157 −0.980785 0.195090i \(-0.937500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(674\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(675\) 0.509796 + 0.509796i 0.509796 + 0.509796i
\(676\) −0.0897902 + 0.216773i −0.0897902 + 0.216773i
\(677\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(678\) −0.0845805 0.126584i −0.0845805 0.126584i
\(679\) 0 0
\(680\) 0 0
\(681\) 0.433546i 0.433546i
\(682\) 0 0
\(683\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(684\) 0.783227 0.324423i 0.783227 0.324423i
\(685\) 0 0
\(686\) 0 0
\(687\) −0.720960 −0.720960
\(688\) 0 0
\(689\) −0.433546 −0.433546
\(690\) 0 0
\(691\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(692\) −0.636379 1.53636i −0.636379 1.53636i
\(693\) 0 0
\(694\) 0 0
\(695\) 1.84776i 1.84776i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(702\) 0.785695 + 0.156284i 0.785695 + 0.156284i
\(703\) −1.96157 −1.96157
\(704\) −0.707107 0.292893i −0.707107 0.292893i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0.601345 0.601345i 0.601345 0.601345i
\(716\) 0 0
\(717\) 0 0
\(718\) 0.360480 1.81225i 0.360480 1.81225i
\(719\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(720\) 0.847759i 0.847759i
\(721\) 0 0
\(722\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0.134381 0.0897902i 0.134381 0.0897902i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0.198912i 0.198912i
\(730\) 0 0
\(731\) 0 0
\(732\) 0.114281 0.275899i 0.114281 0.275899i
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) 0.390181 0.390181
\(736\) 0 0
\(737\) −1.27276 −1.27276
\(738\) 0 0
\(739\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(740\) −0.750661 + 1.81225i −0.750661 + 1.81225i
\(741\) −0.306563 + 0.306563i −0.306563 + 0.306563i
\(742\) 0 0
\(743\) 1.11114i 1.11114i 0.831470 + 0.555570i \(0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(744\) 0 0
\(745\) 1.84776i 1.84776i
\(746\) −0.324423 + 0.216773i −0.324423 + 0.216773i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(761\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(762\) 0.360480 + 0.539496i 0.360480 + 0.539496i
\(763\) 0 0
\(764\) −1.30656 0.541196i −1.30656 0.541196i
\(765\) 0 0
\(766\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(767\) 0 0
\(768\) 0.275899 0.275899i 0.275899 0.275899i
\(769\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(770\) 0 0
\(771\) −0.458804 0.458804i −0.458804 0.458804i
\(772\) 1.53636 + 0.636379i 1.53636 + 0.636379i
\(773\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.923880 + 0.617317i 0.923880 + 0.617317i
\(777\) 0 0
\(778\) 1.17588 0.785695i 1.17588 0.785695i
\(779\) 0 0
\(780\) 0.165911 + 0.400544i 0.165911 + 0.400544i
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.707107 0.707107i −0.707107 0.707107i
\(785\) 0 0
\(786\) −0.107651 + 0.541196i −0.107651 + 0.541196i
\(787\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.126584 + 0.636379i 0.126584 + 0.636379i
\(793\) 0.850430i 0.850430i
\(794\) 0 0
\(795\) −0.107651 + 0.107651i −0.107651 + 0.107651i
\(796\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(797\) −1.38704 1.38704i −1.38704 1.38704i −0.831470 0.555570i \(-0.812500\pi\)
−0.555570 0.831470i \(-0.687500\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.831470 0.555570i 0.831470 0.555570i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.248303 0.599456i 0.248303 0.599456i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.750661 0.149316i 0.750661 0.149316i
\(809\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(810\) −0.470990 + 0.314705i −0.470990 + 0.314705i
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 0.509796 + 0.509796i 0.509796 + 0.509796i
\(814\) 0.292893 1.47247i 0.292893 1.47247i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 1.08979 1.63099i 1.08979 1.63099i
\(825\) 0.298631i 0.298631i
\(826\) 0 0
\(827\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(828\) 0 0
\(829\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.425215 1.02656i 0.425215 1.02656i
\(833\) 0 0
\(834\) 0.707107 + 0.140652i 0.707107 + 0.140652i
\(835\) −1.17588 1.17588i −1.17588 1.17588i
\(836\) −0.707107 0.292893i −0.707107 0.292893i
\(837\) 0 0
\(838\) 0.785695 + 1.17588i 0.785695 + 1.17588i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 1.00000i 1.00000i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.165911 + 0.165911i 0.165911 + 0.165911i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.390181 0.390181
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0.847759i 0.847759i
\(856\) −1.63099 + 0.324423i −1.63099 + 0.324423i
\(857\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(858\) −0.184350 0.275899i −0.184350 0.275899i
\(859\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.66294 −1.66294 −0.831470 0.555570i \(-0.812500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(864\) −0.707107 0.140652i −0.707107 0.140652i
\(865\) −1.66294 −1.66294
\(866\) −0.382683 0.0761205i −0.382683 0.0761205i
\(867\) −0.275899 0.275899i −0.275899 0.275899i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 1.84776i 1.84776i
\(872\) 0 0
\(873\) 0.941979i 0.941979i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.275899 + 0.275899i 0.275899 + 0.275899i 0.831470 0.555570i \(-0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(878\) 0 0
\(879\) −0.765367 −0.765367
\(880\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(881\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(882\) −0.165390 + 0.831470i −0.165390 + 0.831470i
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.390181i 0.390181i −0.980785 0.195090i \(-0.937500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(888\) 0.636379 + 0.425215i 0.636379 + 0.425215i
\(889\) 0 0
\(890\) 0 0
\(891\) 0.306563 0.306563i 0.306563 0.306563i
\(892\) 1.02656 + 0.425215i 1.02656 + 0.425215i
\(893\) 0 0
\(894\) 0.707107 + 0.140652i 0.707107 + 0.140652i
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.783227 0.324423i −0.783227 0.324423i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(908\) −0.425215 1.02656i −0.425215 1.02656i
\(909\) −0.458804 0.458804i −0.458804 0.458804i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0.275899 0.275899i 0.275899 0.275899i
\(913\) 0 0
\(914\) 0 0
\(915\) −0.211164 0.211164i −0.211164 0.211164i
\(916\) 1.70711 0.707107i 1.70711 0.707107i
\(917\) 0 0
\(918\) 0 0
\(919\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(920\) 0 0
\(921\) 0.433546i 0.433546i
\(922\) −0.785695 1.17588i −0.785695 1.17588i
\(923\) 0 0
\(924\) 0 0
\(925\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(926\) 0 0
\(927\) −1.66294 −1.66294
\(928\) 0 0
\(929\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(930\) 0 0
\(931\) −0.707107 0.707107i −0.707107 0.707107i
\(932\) 0 0
\(933\) −0.211164 + 0.211164i −0.211164 + 0.211164i
\(934\) 0 0
\(935\) 0 0
\(936\) −0.923880 + 0.183771i −0.923880 + 0.183771i
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.831470 0.555570i 0.831470 0.555570i
\(951\) 0.648847i 0.648847i
\(952\) 0 0
\(953\) 1.11114i 1.11114i −0.831470 0.555570i \(-0.812500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(954\) −0.183771 0.275033i −0.183771 0.275033i
\(955\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(956\) 0 0
\(957\) 0 0
\(958\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(959\) 0 0
\(960\) −0.149316 0.360480i −0.149316 0.360480i
\(961\) 1.00000 1.00000
\(962\) 2.13770 + 0.425215i 2.13770 + 0.425215i
\(963\) 0.996859 + 0.996859i 0.996859 + 0.996859i
\(964\) 0 0
\(965\) 1.17588 1.17588i 1.17588 1.17588i
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −0.230125 + 0.344406i −0.230125 + 0.344406i
\(969\) 0 0
\(970\) 0.923880 0.617317i 0.923880 0.617317i
\(971\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(972\) 0.360480 + 0.870275i 0.360480 + 0.870275i
\(973\) 0 0
\(974\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i
\(975\) 0.433546 0.433546
\(976\) 0.765367i 0.765367i
\(977\) −1.96157 −1.96157 −0.980785 0.195090i \(-0.937500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(981\) 0 0
\(982\) 1.66294 1.11114i 1.66294 1.11114i
\(983\) 1.96157i 1.96157i 0.195090 + 0.980785i \(0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0.425215 1.02656i 0.425215 1.02656i
\(989\) 0 0
\(990\) 0.636379 + 0.126584i 0.636379 + 0.126584i
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(996\) 0 0
\(997\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(998\) −1.02656 1.53636i −1.02656 1.53636i
\(999\) 1.41421i 1.41421i
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 1520.1.bh.c.189.8 yes 16
5.4 even 2 inner 1520.1.bh.c.189.1 16
16.5 even 4 inner 1520.1.bh.c.949.8 yes 16
19.18 odd 2 inner 1520.1.bh.c.189.1 16
80.69 even 4 inner 1520.1.bh.c.949.1 yes 16
95.94 odd 2 CM 1520.1.bh.c.189.8 yes 16
304.37 odd 4 inner 1520.1.bh.c.949.1 yes 16
1520.949 odd 4 inner 1520.1.bh.c.949.8 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1520.1.bh.c.189.1 16 5.4 even 2 inner
1520.1.bh.c.189.1 16 19.18 odd 2 inner
1520.1.bh.c.189.8 yes 16 1.1 even 1 trivial
1520.1.bh.c.189.8 yes 16 95.94 odd 2 CM
1520.1.bh.c.949.1 yes 16 80.69 even 4 inner
1520.1.bh.c.949.1 yes 16 304.37 odd 4 inner
1520.1.bh.c.949.8 yes 16 16.5 even 4 inner
1520.1.bh.c.949.8 yes 16 1520.949 odd 4 inner