Properties

Label 1020.1.cl.b.719.1
Level $1020$
Weight $1$
Character 1020.719
Analytic conductor $0.509$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1020,1,Mod(299,1020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1020, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 8, 8, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1020.299");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1020.cl (of order \(16\), degree \(8\), 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: \(0.509046312886\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
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 719.1
Root \(0.831470 + 0.555570i\) of defining polynomial
Character \(\chi\) \(=\) 1020.719
Dual form 1020.1.cl.b.959.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.555570 + 0.831470i) q^{2} +(0.831470 - 0.555570i) q^{3} +(-0.382683 - 0.923880i) q^{4} +(-0.195090 + 0.980785i) q^{5} +1.00000i q^{6} +(0.980785 + 0.195090i) q^{8} +(0.382683 - 0.923880i) q^{9} +O(q^{10})\) \(q+(-0.555570 + 0.831470i) q^{2} +(0.831470 - 0.555570i) q^{3} +(-0.382683 - 0.923880i) q^{4} +(-0.195090 + 0.980785i) q^{5} +1.00000i q^{6} +(0.980785 + 0.195090i) q^{8} +(0.382683 - 0.923880i) q^{9} +(-0.707107 - 0.707107i) q^{10} +(-0.831470 - 0.555570i) q^{12} +(0.382683 + 0.923880i) q^{15} +(-0.707107 + 0.707107i) q^{16} +(0.555570 + 0.831470i) q^{17} +(0.555570 + 0.831470i) q^{18} +(0.707107 - 0.292893i) q^{19} +(0.980785 - 0.195090i) q^{20} +(1.17588 + 0.785695i) q^{23} +(0.923880 - 0.382683i) q^{24} +(-0.923880 - 0.382683i) q^{25} +(-0.195090 - 0.980785i) q^{27} +(-0.980785 - 0.195090i) q^{30} +(-0.216773 - 0.324423i) q^{31} +(-0.195090 - 0.980785i) q^{32} -1.00000 q^{34} -1.00000 q^{36} +(-0.149316 + 0.750661i) q^{38} +(-0.382683 + 0.923880i) q^{40} +(0.831470 + 0.555570i) q^{45} +(-1.30656 + 0.541196i) q^{46} +(-1.38704 + 1.38704i) q^{47} +(-0.195090 + 0.980785i) q^{48} +(0.923880 - 0.382683i) q^{49} +(0.831470 - 0.555570i) q^{50} +(0.923880 + 0.382683i) q^{51} +(-0.636379 - 1.53636i) q^{53} +(0.923880 + 0.382683i) q^{54} +(0.425215 - 0.636379i) q^{57} +(0.707107 - 0.707107i) q^{60} +(0.382683 - 0.0761205i) q^{61} +0.390181 q^{62} +(0.923880 + 0.382683i) q^{64} +(0.555570 - 0.831470i) q^{68} +1.41421 q^{69} +(0.555570 - 0.831470i) q^{72} +(-0.980785 + 0.195090i) q^{75} +(-0.541196 - 0.541196i) q^{76} +(-1.08979 + 1.63099i) q^{79} +(-0.555570 - 0.831470i) q^{80} +(-0.707107 - 0.707107i) q^{81} +(-0.425215 - 1.02656i) q^{83} +(-0.923880 + 0.382683i) q^{85} +(-0.923880 + 0.382683i) q^{90} +(0.275899 - 1.38704i) q^{92} +(-0.360480 - 0.149316i) q^{93} +(-0.382683 - 1.92388i) q^{94} +(0.149316 + 0.750661i) q^{95} +(-0.707107 - 0.707107i) q^{96} +(-0.195090 + 0.980785i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{34} - 16 q^{36}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(e\left(\frac{5}{16}\right)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(3\) 0.831470 0.555570i 0.831470 0.555570i
\(4\) −0.382683 0.923880i −0.382683 0.923880i
\(5\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(6\) 1.00000i 1.00000i
\(7\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(8\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(9\) 0.382683 0.923880i 0.382683 0.923880i
\(10\) −0.707107 0.707107i −0.707107 0.707107i
\(11\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(12\) −0.831470 0.555570i −0.831470 0.555570i
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(16\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(17\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(18\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(19\) 0.707107 0.292893i 0.707107 0.292893i 1.00000i \(-0.5\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 0.980785 0.195090i 0.980785 0.195090i
\(21\) 0 0
\(22\) 0 0
\(23\) 1.17588 + 0.785695i 1.17588 + 0.785695i 0.980785 0.195090i \(-0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(24\) 0.923880 0.382683i 0.923880 0.382683i
\(25\) −0.923880 0.382683i −0.923880 0.382683i
\(26\) 0 0
\(27\) −0.195090 0.980785i −0.195090 0.980785i
\(28\) 0 0
\(29\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(30\) −0.980785 0.195090i −0.980785 0.195090i
\(31\) −0.216773 0.324423i −0.216773 0.324423i 0.707107 0.707107i \(-0.250000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(32\) −0.195090 0.980785i −0.195090 0.980785i
\(33\) 0 0
\(34\) −1.00000 −1.00000
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(38\) −0.149316 + 0.750661i −0.149316 + 0.750661i
\(39\) 0 0
\(40\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(41\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(42\) 0 0
\(43\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(44\) 0 0
\(45\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(46\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(47\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(48\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(49\) 0.923880 0.382683i 0.923880 0.382683i
\(50\) 0.831470 0.555570i 0.831470 0.555570i
\(51\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(52\) 0 0
\(53\) −0.636379 1.53636i −0.636379 1.53636i −0.831470 0.555570i \(-0.812500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(54\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.425215 0.636379i 0.425215 0.636379i
\(58\) 0 0
\(59\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(60\) 0.707107 0.707107i 0.707107 0.707107i
\(61\) 0.382683 0.0761205i 0.382683 0.0761205i 1.00000i \(-0.5\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(62\) 0.390181 0.390181
\(63\) 0 0
\(64\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0.555570 0.831470i 0.555570 0.831470i
\(69\) 1.41421 1.41421
\(70\) 0 0
\(71\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(72\) 0.555570 0.831470i 0.555570 0.831470i
\(73\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(74\) 0 0
\(75\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(76\) −0.541196 0.541196i −0.541196 0.541196i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.08979 + 1.63099i −1.08979 + 1.63099i −0.382683 + 0.923880i \(0.625000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) −0.555570 0.831470i −0.555570 0.831470i
\(81\) −0.707107 0.707107i −0.707107 0.707107i
\(82\) 0 0
\(83\) −0.425215 1.02656i −0.425215 1.02656i −0.980785 0.195090i \(-0.937500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(84\) 0 0
\(85\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(91\) 0 0
\(92\) 0.275899 1.38704i 0.275899 1.38704i
\(93\) −0.360480 0.149316i −0.360480 0.149316i
\(94\) −0.382683 1.92388i −0.382683 1.92388i
\(95\) 0.149316 + 0.750661i 0.149316 + 0.750661i
\(96\) −0.707107 0.707107i −0.707107 0.707107i
\(97\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(98\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(99\) 0 0
\(100\) 1.00000i 1.00000i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(107\) −1.81225 0.360480i −1.81225 0.360480i −0.831470 0.555570i \(-0.812500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(108\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(109\) 0.216773 + 1.08979i 0.216773 + 1.08979i 0.923880 + 0.382683i \(0.125000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.53636 1.02656i −1.53636 1.02656i −0.980785 0.195090i \(-0.937500\pi\)
−0.555570 0.831470i \(-0.687500\pi\)
\(114\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(115\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(121\) −0.382683 0.923880i −0.382683 0.923880i
\(122\) −0.149316 + 0.360480i −0.149316 + 0.360480i
\(123\) 0 0
\(124\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(125\) 0.555570 0.831470i 0.555570 0.831470i
\(126\) 0 0
\(127\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(128\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 1.00000
\(136\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(137\) −1.96157 −1.96157 −0.980785 0.195090i \(-0.937500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(138\) −0.785695 + 1.17588i −0.785695 + 1.17588i
\(139\) 1.38268 0.923880i 1.38268 0.923880i 0.382683 0.923880i \(-0.375000\pi\)
1.00000 \(0\)
\(140\) 0 0
\(141\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(142\) 0 0
\(143\) 0 0
\(144\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.555570 0.831470i 0.555570 0.831470i
\(148\) 0 0
\(149\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(150\) 0.382683 0.923880i 0.382683 0.923880i
\(151\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(152\) 0.750661 0.149316i 0.750661 0.149316i
\(153\) 0.980785 0.195090i 0.980785 0.195090i
\(154\) 0 0
\(155\) 0.360480 0.149316i 0.360480 0.149316i
\(156\) 0 0
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) −0.750661 1.81225i −0.750661 1.81225i
\(159\) −1.38268 0.923880i −1.38268 0.923880i
\(160\) 1.00000 1.00000
\(161\) 0 0
\(162\) 0.980785 0.195090i 0.980785 0.195090i
\(163\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(167\) 1.02656 + 1.53636i 1.02656 + 1.53636i 0.831470 + 0.555570i \(0.187500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 0.195090 0.980785i 0.195090 0.980785i
\(171\) 0.765367i 0.765367i
\(172\) 0 0
\(173\) −1.02656 1.53636i −1.02656 1.53636i −0.831470 0.555570i \(-0.812500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(180\) 0.195090 0.980785i 0.195090 0.980785i
\(181\) −1.38268 0.923880i −1.38268 0.923880i −0.382683 0.923880i \(-0.625000\pi\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0.275899 0.275899i 0.275899 0.275899i
\(184\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(185\) 0 0
\(186\) 0.324423 0.216773i 0.324423 0.216773i
\(187\) 0 0
\(188\) 1.81225 + 0.750661i 1.81225 + 0.750661i
\(189\) 0 0
\(190\) −0.707107 0.292893i −0.707107 0.292893i
\(191\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(192\) 0.980785 0.195090i 0.980785 0.195090i
\(193\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.707107 0.707107i −0.707107 0.707107i
\(197\) 1.38704 0.275899i 1.38704 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(198\) 0 0
\(199\) 0.216773 1.08979i 0.216773 1.08979i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(200\) −0.831470 0.555570i −0.831470 0.555570i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 1.00000i 1.00000i
\(205\) 0 0
\(206\) 0 0
\(207\) 1.17588 0.785695i 1.17588 0.785695i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.08979 + 0.216773i −1.08979 + 0.216773i −0.707107 0.707107i \(-0.750000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(212\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(213\) 0 0
\(214\) 1.30656 1.30656i 1.30656 1.30656i
\(215\) 0 0
\(216\) 1.00000i 1.00000i
\(217\) 0 0
\(218\) −1.02656 0.425215i −1.02656 0.425215i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(224\) 0 0
\(225\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(226\) 1.70711 0.707107i 1.70711 0.707107i
\(227\) 0.636379 + 0.425215i 0.636379 + 0.425215i 0.831470 0.555570i \(-0.187500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(228\) −0.750661 0.149316i −0.750661 0.149316i
\(229\) −1.70711 0.707107i −1.70711 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
−1.00000 \(\pi\)
\(230\) −0.275899 1.38704i −0.275899 1.38704i
\(231\) 0 0
\(232\) 0 0
\(233\) 1.38704 + 0.275899i 1.38704 + 0.275899i 0.831470 0.555570i \(-0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(234\) 0 0
\(235\) −1.08979 1.63099i −1.08979 1.63099i
\(236\) 0 0
\(237\) 1.96157i 1.96157i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −0.923880 0.382683i −0.923880 0.382683i
\(241\) 1.08979 + 1.63099i 1.08979 + 1.63099i 0.707107 + 0.707107i \(0.250000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(242\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(243\) −0.980785 0.195090i −0.980785 0.195090i
\(244\) −0.216773 0.324423i −0.216773 0.324423i
\(245\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(246\) 0 0
\(247\) 0 0
\(248\) −0.149316 0.360480i −0.149316 0.360480i
\(249\) −0.923880 0.617317i −0.923880 0.617317i
\(250\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(251\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(256\) 1.00000i 1.00000i
\(257\) 0.149316 + 0.360480i 0.149316 + 0.360480i 0.980785 0.195090i \(-0.0625000\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.425215 1.02656i 0.425215 1.02656i −0.555570 0.831470i \(-0.687500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(264\) 0 0
\(265\) 1.63099 0.324423i 1.63099 0.324423i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(270\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(271\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(272\) −0.980785 0.195090i −0.980785 0.195090i
\(273\) 0 0
\(274\) 1.08979 1.63099i 1.08979 1.63099i
\(275\) 0 0
\(276\) −0.541196 1.30656i −0.541196 1.30656i
\(277\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(278\) 1.66294i 1.66294i
\(279\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(280\) 0 0
\(281\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(282\) −1.38704 1.38704i −1.38704 1.38704i
\(283\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(284\) 0 0
\(285\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(286\) 0 0
\(287\) 0 0
\(288\) −0.980785 0.195090i −0.980785 0.195090i
\(289\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(294\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(301\) 0 0
\(302\) 1.38704 + 0.275899i 1.38704 + 0.275899i
\(303\) 0 0
\(304\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(305\) 0.390181i 0.390181i
\(306\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.0761205 + 0.382683i −0.0761205 + 0.382683i
\(311\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(317\) 0.636379 + 0.425215i 0.636379 + 0.425215i 0.831470 0.555570i \(-0.187500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(318\) 1.53636 0.636379i 1.53636 0.636379i
\(319\) 0 0
\(320\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(321\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(322\) 0 0
\(323\) 0.636379 + 0.425215i 0.636379 + 0.425215i
\(324\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(325\) 0 0
\(326\) 0 0
\(327\) 0.785695 + 0.785695i 0.785695 + 0.785695i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.707107 + 1.70711i −0.707107 + 1.70711i 1.00000i \(0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) −0.785695 + 0.785695i −0.785695 + 0.785695i
\(333\) 0 0
\(334\) −1.84776 −1.84776
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(338\) −0.831470 0.555570i −0.831470 0.555570i
\(339\) −1.84776 −1.84776
\(340\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(341\) 0 0
\(342\) 0.636379 + 0.425215i 0.636379 + 0.425215i
\(343\) 0 0
\(344\) 0 0
\(345\) −0.275899 + 1.38704i −0.275899 + 1.38704i
\(346\) 1.84776 1.84776
\(347\) 0.750661 0.149316i 0.750661 0.149316i 0.195090 0.980785i \(-0.437500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(348\) 0 0
\(349\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.17588 + 1.17588i 1.17588 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(360\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(361\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(362\) 1.53636 0.636379i 1.53636 0.636379i
\(363\) −0.831470 0.555570i −0.831470 0.555570i
\(364\) 0 0
\(365\) 0 0
\(366\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(367\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(368\) −1.38704 + 0.275899i −1.38704 + 0.275899i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.390181i 0.390181i
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 1.00000i 1.00000i
\(376\) −1.63099 + 1.08979i −1.63099 + 1.08979i
\(377\) 0 0
\(378\) 0 0
\(379\) 1.08979 + 0.216773i 1.08979 + 0.216773i 0.707107 0.707107i \(-0.250000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(380\) 0.636379 0.425215i 0.636379 0.425215i
\(381\) 0 0
\(382\) 0 0
\(383\) −0.360480 0.149316i −0.360480 0.149316i 0.195090 0.980785i \(-0.437500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(384\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(390\) 0 0
\(391\) 1.41421i 1.41421i
\(392\) 0.980785 0.195090i 0.980785 0.195090i
\(393\) 0 0
\(394\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(395\) −1.38704 1.38704i −1.38704 1.38704i
\(396\) 0 0
\(397\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(398\) 0.785695 + 0.785695i 0.785695 + 0.785695i
\(399\) 0 0
\(400\) 0.923880 0.382683i 0.923880 0.382683i
\(401\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.831470 0.555570i 0.831470 0.555570i
\(406\) 0 0
\(407\) 0 0
\(408\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(409\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(410\) 0 0
\(411\) −1.63099 + 1.08979i −1.63099 + 1.08979i
\(412\) 0 0
\(413\) 0 0
\(414\) 1.41421i 1.41421i
\(415\) 1.08979 0.216773i 1.08979 0.216773i
\(416\) 0 0
\(417\) 0.636379 1.53636i 0.636379 1.53636i
\(418\) 0 0
\(419\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(420\) 0 0
\(421\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(422\) 0.425215 1.02656i 0.425215 1.02656i
\(423\) 0.750661 + 1.81225i 0.750661 + 1.81225i
\(424\) −0.324423 1.63099i −0.324423 1.63099i
\(425\) −0.195090 0.980785i −0.195090 0.980785i
\(426\) 0 0
\(427\) 0 0
\(428\) 0.360480 + 1.81225i 0.360480 + 1.81225i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(432\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(433\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.923880 0.617317i 0.923880 0.617317i
\(437\) 1.06159 + 0.211164i 1.06159 + 0.211164i
\(438\) 0 0
\(439\) 0.923880 + 1.38268i 0.923880 + 1.38268i 0.923880 + 0.382683i \(0.125000\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 1.00000i 1.00000i
\(442\) 0 0
\(443\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(450\) −0.195090 0.980785i −0.195090 0.980785i
\(451\) 0 0
\(452\) −0.360480 + 1.81225i −0.360480 + 1.81225i
\(453\) −1.17588 0.785695i −1.17588 0.785695i
\(454\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(455\) 0 0
\(456\) 0.541196 0.541196i 0.541196 0.541196i
\(457\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(458\) 1.53636 1.02656i 1.53636 1.02656i
\(459\) 0.707107 0.707107i 0.707107 0.707107i
\(460\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(461\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0.216773 0.324423i 0.216773 0.324423i
\(466\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(467\) −0.149316 + 0.360480i −0.149316 + 0.360480i −0.980785 0.195090i \(-0.937500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.96157 1.96157
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) −1.63099 1.08979i −1.63099 1.08979i
\(475\) −0.765367 −0.765367
\(476\) 0 0
\(477\) −1.66294 −1.66294
\(478\) 0 0
\(479\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(480\) 0.831470 0.555570i 0.831470 0.555570i
\(481\) 0 0
\(482\) −1.96157 −1.96157
\(483\) 0 0
\(484\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(485\) 0 0
\(486\) 0.707107 0.707107i 0.707107 0.707107i
\(487\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(488\) 0.390181 0.390181
\(489\) 0 0
\(490\) −0.923880 0.382683i −0.923880 0.382683i
\(491\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(497\) 0 0
\(498\) 1.02656 0.425215i 1.02656 0.425215i
\(499\) 1.38268 + 0.923880i 1.38268 + 0.923880i 1.00000 \(0\)
0.382683 + 0.923880i \(0.375000\pi\)
\(500\) −0.980785 0.195090i −0.980785 0.195090i
\(501\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(502\) 0 0
\(503\) 0.360480 + 1.81225i 0.360480 + 1.81225i 0.555570 + 0.831470i \(0.312500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) −0.382683 0.923880i −0.382683 0.923880i
\(511\) 0 0
\(512\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(513\) −0.425215 0.636379i −0.425215 0.636379i
\(514\) −0.382683 0.0761205i −0.382683 0.0761205i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.70711 0.707107i −1.70711 0.707107i
\(520\) 0 0
\(521\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(527\) 0.149316 0.360480i 0.149316 0.360480i
\(528\) 0 0
\(529\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(530\) −0.636379 + 1.53636i −0.636379 + 1.53636i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0.707107 1.70711i 0.707107 1.70711i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −0.382683 0.923880i −0.382683 0.923880i
\(541\) 1.38268 0.923880i 1.38268 0.923880i 0.382683 0.923880i \(-0.375000\pi\)
1.00000 \(0\)
\(542\) −1.02656 + 1.53636i −1.02656 + 1.53636i
\(543\) −1.66294 −1.66294
\(544\) 0.707107 0.707107i 0.707107 0.707107i
\(545\) −1.11114 −1.11114
\(546\) 0 0
\(547\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(548\) 0.750661 + 1.81225i 0.750661 + 1.81225i
\(549\) 0.0761205 0.382683i 0.0761205 0.382683i
\(550\) 0 0
\(551\) 0 0
\(552\) 1.38704 + 0.275899i 1.38704 + 0.275899i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.38268 0.923880i −1.38268 0.923880i
\(557\) −0.785695 0.785695i −0.785695 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(558\) 0.149316 0.360480i 0.149316 0.360480i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.53636 0.636379i 1.53636 0.636379i 0.555570 0.831470i \(-0.312500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(564\) 1.92388 0.382683i 1.92388 0.382683i
\(565\) 1.30656 1.30656i 1.30656 1.30656i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(570\) −0.750661 + 0.149316i −0.750661 + 0.149316i
\(571\) −0.0761205 0.382683i −0.0761205 0.382683i 0.923880 0.382683i \(-0.125000\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.785695 1.17588i −0.785695 1.17588i
\(576\) 0.707107 0.707107i 0.707107 0.707107i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −0.555570 0.831470i −0.555570 0.831470i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −0.382683 1.92388i −0.382683 1.92388i
\(587\) −1.53636 0.636379i −1.53636 0.636379i −0.555570 0.831470i \(-0.687500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(588\) −0.980785 0.195090i −0.980785 0.195090i
\(589\) −0.248303 0.165911i −0.248303 0.165911i
\(590\) 0 0
\(591\) 1.00000 1.00000i 1.00000 1.00000i
\(592\) 0 0
\(593\) 1.81225 0.750661i 1.81225 0.750661i 0.831470 0.555570i \(-0.187500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.425215 1.02656i −0.425215 1.02656i
\(598\) 0 0
\(599\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(600\) −1.00000 −1.00000
\(601\) −0.216773 + 0.324423i −0.216773 + 0.324423i −0.923880 0.382683i \(-0.875000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(605\) 0.980785 0.195090i 0.980785 0.195090i
\(606\) 0 0
\(607\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(608\) −0.425215 0.636379i −0.425215 0.636379i
\(609\) 0 0
\(610\) −0.324423 0.216773i −0.324423 0.216773i
\(611\) 0 0
\(612\) −0.555570 0.831470i −0.555570 0.831470i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.360480 + 1.81225i −0.360480 + 1.81225i 0.195090 + 0.980785i \(0.437500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(618\) 0 0
\(619\) 0.382683 0.0761205i 0.382683 0.0761205i 1.00000i \(-0.5\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(620\) −0.275899 0.275899i −0.275899 0.275899i
\(621\) 0.541196 1.30656i 0.541196 1.30656i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(632\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(633\) −0.785695 + 0.785695i −0.785695 + 0.785695i
\(634\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(635\) 0 0
\(636\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −0.382683 0.923880i −0.382683 0.923880i
\(641\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(642\) 0.360480 1.81225i 0.360480 1.81225i
\(643\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(647\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(648\) −0.555570 0.831470i −0.555570 0.831470i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.149316 + 0.750661i 0.149316 + 0.750661i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(654\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) −0.707107 + 0.292893i −0.707107 + 0.292893i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(0.5\pi\)
\(662\) −1.02656 1.53636i −1.02656 1.53636i
\(663\) 0 0
\(664\) −0.216773 1.08979i −0.216773 1.08979i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.02656 1.53636i 1.02656 1.53636i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(674\) 0 0
\(675\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(676\) 0.923880 0.382683i 0.923880 0.382683i
\(677\) 1.17588 0.785695i 1.17588 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(678\) 1.02656 1.53636i 1.02656 1.53636i
\(679\) 0 0
\(680\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(681\) 0.765367 0.765367
\(682\) 0 0
\(683\) −1.53636 + 1.02656i −1.53636 + 1.02656i −0.555570 + 0.831470i \(0.687500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(684\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(685\) 0.382683 1.92388i 0.382683 1.92388i
\(686\) 0 0
\(687\) −1.81225 + 0.360480i −1.81225 + 0.360480i
\(688\) 0 0
\(689\) 0 0
\(690\) −1.00000 1.00000i −1.00000 1.00000i
\(691\) 1.08979 1.63099i 1.08979 1.63099i 0.382683 0.923880i \(-0.375000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(692\) −1.02656 + 1.53636i −1.02656 + 1.53636i
\(693\) 0 0
\(694\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(695\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 1.30656 0.541196i 1.30656 0.541196i
\(700\) 0 0
\(701\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −1.81225 0.750661i −1.81225 0.750661i
\(706\) −1.63099 + 0.324423i −1.63099 + 0.324423i
\(707\) 0 0
\(708\) 0 0
\(709\) −1.08979 0.216773i −1.08979 0.216773i −0.382683 0.923880i \(-0.625000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(710\) 0 0
\(711\) 1.08979 + 1.63099i 1.08979 + 1.63099i
\(712\) 0 0
\(713\) 0.551799i 0.551799i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(720\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(721\) 0 0
\(722\) −0.0808091 0.406255i −0.0808091 0.406255i
\(723\) 1.81225 + 0.750661i 1.81225 + 0.750661i
\(724\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(725\) 0 0
\(726\) 0.923880 0.382683i 0.923880 0.382683i
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(730\) 0 0
\(731\) 0 0
\(732\) −0.360480 0.149316i −0.360480 0.149316i
\(733\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(734\) 0 0
\(735\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(736\) 0.541196 1.30656i 0.541196 1.30656i
\(737\) 0 0
\(738\) 0 0
\(739\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.149316 + 0.750661i −0.149316 + 0.750661i 0.831470 + 0.555570i \(0.187500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(744\) −0.324423 0.216773i −0.324423 0.216773i
\(745\) 0 0
\(746\) 0 0
\(747\) −1.11114 −1.11114
\(748\) 0 0
\(749\) 0 0
\(750\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(751\) 0.923880 0.617317i 0.923880 0.617317i 1.00000i \(-0.5\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(752\) 1.96157i 1.96157i
\(753\) 0 0
\(754\) 0 0
\(755\) 1.38704 0.275899i 1.38704 0.275899i
\(756\) 0 0
\(757\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(758\) −0.785695 + 0.785695i −0.785695 + 0.785695i
\(759\) 0 0
\(760\) 0.765367i 0.765367i
\(761\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.00000i 1.00000i
\(766\) 0.324423 0.216773i 0.324423 0.216773i
\(767\) 0 0
\(768\) −0.555570 0.831470i −0.555570 0.831470i
\(769\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(772\) 0 0
\(773\) 1.02656 + 0.425215i 1.02656 + 0.425215i 0.831470 0.555570i \(-0.187500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(774\) 0 0
\(775\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −1.17588 0.785695i −1.17588 0.785695i
\(783\) 0 0
\(784\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(788\) −0.785695 1.17588i −0.785695 1.17588i
\(789\) −0.216773 1.08979i −0.216773 1.08979i
\(790\) 1.92388 0.382683i 1.92388 0.382683i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1.17588 1.17588i 1.17588 1.17588i
\(796\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(797\) −0.360480 + 0.149316i −0.360480 + 0.149316i −0.555570 0.831470i \(-0.687500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(798\) 0 0
\(799\) −1.92388 0.382683i −1.92388 0.382683i
\(800\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(801\) 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 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(810\) 1.00000i 1.00000i
\(811\) −0.216773 + 1.08979i −0.216773 + 1.08979i 0.707107 + 0.707107i \(0.250000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(812\) 0 0
\(813\) 1.53636 1.02656i 1.53636 1.02656i
\(814\) 0 0
\(815\) 0 0
\(816\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(817\) 0 0
\(818\) 0.425215 0.636379i 0.425215 0.636379i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(822\) 1.96157i 1.96157i
\(823\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.785695 1.17588i 0.785695 1.17588i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(828\) −1.17588 0.785695i −1.17588 0.785695i
\(829\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(830\) −0.425215 + 1.02656i −0.425215 + 1.02656i
\(831\) 0 0
\(832\) 0 0
\(833\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(834\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(835\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(836\) 0 0
\(837\) −0.275899 + 0.275899i −0.275899 + 0.275899i
\(838\) 0 0
\(839\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(840\) 0 0
\(841\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(842\) 0.750661 0.149316i 0.750661 0.149316i
\(843\) 0 0
\(844\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(845\) −0.980785 0.195090i −0.980785 0.195090i
\(846\) −1.92388 0.382683i −1.92388 0.382683i
\(847\) 0 0
\(848\) 1.53636 + 0.636379i 1.53636 + 0.636379i
\(849\) 0 0
\(850\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(854\) 0 0
\(855\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(856\) −1.70711 0.707107i −1.70711 0.707107i
\(857\) −0.275899 1.38704i −0.275899 1.38704i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(858\) 0 0
\(859\) 1.70711 + 0.707107i 1.70711 + 0.707107i 1.00000 \(0\)
0.707107 + 0.707107i \(0.250000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(864\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(865\) 1.70711 0.707107i 1.70711 0.707107i
\(866\) 0 0
\(867\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.11114i 1.11114i
\(873\) 0 0
\(874\) −0.765367 + 0.765367i −0.765367 + 0.765367i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(878\) −1.66294 −1.66294
\(879\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(880\) 0 0
\(881\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(882\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(887\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.574531 + 1.38704i −0.574531 + 1.38704i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(901\) 0.923880 1.38268i 0.923880 1.38268i
\(902\) 0 0
\(903\) 0 0
\(904\) −1.30656 1.30656i −1.30656 1.30656i
\(905\) 1.17588 1.17588i 1.17588 1.17588i
\(906\) 1.30656 0.541196i 1.30656 0.541196i
\(907\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(908\) 0.149316 0.750661i 0.149316 0.750661i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(912\) 0.149316 + 0.750661i 0.149316 + 0.750661i
\(913\) 0 0
\(914\) 0 0
\(915\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(916\) 1.84776i 1.84776i
\(917\) 0 0
\(918\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(919\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(920\) −1.17588 + 0.785695i −1.17588 + 0.785695i
\(921\) 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 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(930\) 0.149316 + 0.360480i 0.149316 + 0.360480i
\(931\) 0.541196 0.541196i 0.541196 0.541196i
\(932\) −0.275899 1.38704i −0.275899 1.38704i
\(933\) 0 0
\(934\) −0.216773 0.324423i −0.216773 0.324423i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.08979 + 1.63099i −1.08979 + 1.63099i
\(941\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(948\) 1.81225 0.750661i 1.81225 0.750661i
\(949\) 0 0
\(950\) 0.425215 0.636379i 0.425215 0.636379i
\(951\) 0.765367 0.765367
\(952\) 0 0
\(953\) −0.390181 −0.390181 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(954\) 0.923880 1.38268i 0.923880 1.38268i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.00000i 1.00000i
\(961\) 0.324423 0.783227i 0.324423 0.783227i
\(962\) 0 0
\(963\) −1.02656 + 1.53636i −1.02656 + 1.53636i
\(964\) 1.08979 1.63099i 1.08979 1.63099i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(968\) −0.195090 0.980785i −0.195090 0.980785i
\(969\) 0.765367 0.765367
\(970\) 0 0
\(971\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(972\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(977\) −1.02656 0.425215i −1.02656 0.425215i −0.195090 0.980785i \(-0.562500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.831470 0.555570i 0.831470 0.555570i
\(981\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(982\) 0 0
\(983\) −0.425215 0.636379i −0.425215 0.636379i 0.555570 0.831470i \(-0.312500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(984\) 0 0
\(985\) 1.41421i 1.41421i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.63099 0.324423i −1.63099 0.324423i −0.707107 0.707107i \(-0.750000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(992\) −0.275899 + 0.275899i −0.275899 + 0.275899i
\(993\) 0.360480 + 1.81225i 0.360480 + 1.81225i
\(994\) 0 0
\(995\) 1.02656 + 0.425215i 1.02656 + 0.425215i
\(996\) −0.216773 + 1.08979i −0.216773 + 1.08979i
\(997\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(998\) −1.53636 + 0.636379i −1.53636 + 0.636379i
\(999\) 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 1020.1.cl.b.719.1 yes 16
3.2 odd 2 inner 1020.1.cl.b.719.2 yes 16
4.3 odd 2 1020.1.cl.a.719.2 yes 16
5.4 even 2 inner 1020.1.cl.b.719.2 yes 16
12.11 even 2 1020.1.cl.a.719.1 16
15.14 odd 2 CM 1020.1.cl.b.719.1 yes 16
17.7 odd 16 1020.1.cl.a.959.2 yes 16
20.19 odd 2 1020.1.cl.a.719.1 16
51.41 even 16 1020.1.cl.a.959.1 yes 16
60.59 even 2 1020.1.cl.a.719.2 yes 16
68.7 even 16 inner 1020.1.cl.b.959.1 yes 16
85.24 odd 16 1020.1.cl.a.959.1 yes 16
204.143 odd 16 inner 1020.1.cl.b.959.2 yes 16
255.194 even 16 1020.1.cl.a.959.2 yes 16
340.279 even 16 inner 1020.1.cl.b.959.2 yes 16
1020.959 odd 16 inner 1020.1.cl.b.959.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1020.1.cl.a.719.1 16 12.11 even 2
1020.1.cl.a.719.1 16 20.19 odd 2
1020.1.cl.a.719.2 yes 16 4.3 odd 2
1020.1.cl.a.719.2 yes 16 60.59 even 2
1020.1.cl.a.959.1 yes 16 51.41 even 16
1020.1.cl.a.959.1 yes 16 85.24 odd 16
1020.1.cl.a.959.2 yes 16 17.7 odd 16
1020.1.cl.a.959.2 yes 16 255.194 even 16
1020.1.cl.b.719.1 yes 16 1.1 even 1 trivial
1020.1.cl.b.719.1 yes 16 15.14 odd 2 CM
1020.1.cl.b.719.2 yes 16 3.2 odd 2 inner
1020.1.cl.b.719.2 yes 16 5.4 even 2 inner
1020.1.cl.b.959.1 yes 16 68.7 even 16 inner
1020.1.cl.b.959.1 yes 16 1020.959 odd 16 inner
1020.1.cl.b.959.2 yes 16 204.143 odd 16 inner
1020.1.cl.b.959.2 yes 16 340.279 even 16 inner