Properties

Label 1520.1.bh.c.189.5
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 189.5
Root \(0.831470 + 0.555570i\) of defining polynomial
Character \(\chi\) \(=\) 1520.189
Dual form 1520.1.bh.c.949.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.195090 - 0.980785i) q^{2} +(1.38704 + 1.38704i) q^{3} +(-0.923880 - 0.382683i) q^{4} +(0.707107 - 0.707107i) q^{5} +(1.63099 - 1.08979i) q^{6} +(-0.555570 + 0.831470i) q^{8} +2.84776i q^{9} +O(q^{10})\) \(q+(0.195090 - 0.980785i) q^{2} +(1.38704 + 1.38704i) q^{3} +(-0.923880 - 0.382683i) q^{4} +(0.707107 - 0.707107i) q^{5} +(1.63099 - 1.08979i) q^{6} +(-0.555570 + 0.831470i) q^{8} +2.84776i q^{9} +(-0.555570 - 0.831470i) q^{10} +(0.541196 - 0.541196i) q^{11} +(-0.750661 - 1.81225i) q^{12} +(-1.17588 - 1.17588i) q^{13} +1.96157 q^{15} +(0.707107 + 0.707107i) q^{16} +(2.79304 + 0.555570i) q^{18} +(0.707107 + 0.707107i) q^{19} +(-0.923880 + 0.382683i) q^{20} +(-0.425215 - 0.636379i) q^{22} +(-1.92388 + 0.382683i) q^{24} -1.00000i q^{25} +(-1.38268 + 0.923880i) q^{26} +(-2.56292 + 2.56292i) q^{27} +(0.382683 - 1.92388i) q^{30} +(0.831470 - 0.555570i) q^{32} +1.50132 q^{33} +(1.08979 - 2.63099i) q^{36} +(-0.275899 + 0.275899i) q^{37} +(0.831470 - 0.555570i) q^{38} -3.26197i q^{39} +(0.195090 + 0.980785i) q^{40} +(-0.707107 + 0.292893i) q^{44} +(2.01367 + 2.01367i) q^{45} +1.96157i q^{48} -1.00000 q^{49} +(-0.980785 - 0.195090i) q^{50} +(0.636379 + 1.53636i) q^{52} +(-1.38704 + 1.38704i) q^{53} +(2.01367 + 3.01367i) q^{54} -0.765367i q^{55} +1.96157i q^{57} +(-1.81225 - 0.750661i) q^{60} +(-0.541196 - 0.541196i) q^{61} +(-0.382683 - 0.923880i) q^{64} -1.66294 q^{65} +(0.292893 - 1.47247i) q^{66} +(-0.785695 - 0.785695i) q^{67} +(-2.36783 - 1.58213i) q^{72} +(0.216773 + 0.324423i) q^{74} +(1.38704 - 1.38704i) q^{75} +(-0.382683 - 0.923880i) q^{76} +(-3.19929 - 0.636379i) q^{78} +1.00000 q^{80} -4.26197 q^{81} +(0.149316 + 0.750661i) q^{88} +(2.36783 - 1.58213i) q^{90} +1.00000 q^{95} +(1.92388 + 0.382683i) q^{96} +1.66294 q^{97} +(-0.195090 + 0.980785i) q^{98} +(1.54120 + 1.54120i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.195090 0.980785i 0.195090 0.980785i
\(3\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(4\) −0.923880 0.382683i −0.923880 0.382683i
\(5\) 0.707107 0.707107i 0.707107 0.707107i
\(6\) 1.63099 1.08979i 1.63099 1.08979i
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(9\) 2.84776i 2.84776i
\(10\) −0.555570 0.831470i −0.555570 0.831470i
\(11\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(12\) −0.750661 1.81225i −0.750661 1.81225i
\(13\) −1.17588 1.17588i −1.17588 1.17588i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(14\) 0 0
\(15\) 1.96157 1.96157
\(16\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 2.79304 + 0.555570i 2.79304 + 0.555570i
\(19\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(20\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(21\) 0 0
\(22\) −0.425215 0.636379i −0.425215 0.636379i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(25\) 1.00000i 1.00000i
\(26\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(27\) −2.56292 + 2.56292i −2.56292 + 2.56292i
\(28\) 0 0
\(29\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(30\) 0.382683 1.92388i 0.382683 1.92388i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.831470 0.555570i 0.831470 0.555570i
\(33\) 1.50132 1.50132
\(34\) 0 0
\(35\) 0 0
\(36\) 1.08979 2.63099i 1.08979 2.63099i
\(37\) −0.275899 + 0.275899i −0.275899 + 0.275899i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(38\) 0.831470 0.555570i 0.831470 0.555570i
\(39\) 3.26197i 3.26197i
\(40\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(41\) 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
\(45\) 2.01367 + 2.01367i 2.01367 + 2.01367i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.96157i 1.96157i
\(49\) −1.00000 −1.00000
\(50\) −0.980785 0.195090i −0.980785 0.195090i
\(51\) 0 0
\(52\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(53\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(54\) 2.01367 + 3.01367i 2.01367 + 3.01367i
\(55\) 0.765367i 0.765367i
\(56\) 0 0
\(57\) 1.96157i 1.96157i
\(58\) 0 0
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) −1.81225 0.750661i −1.81225 0.750661i
\(61\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.382683 0.923880i −0.382683 0.923880i
\(65\) −1.66294 −1.66294
\(66\) 0.292893 1.47247i 0.292893 1.47247i
\(67\) −0.785695 0.785695i −0.785695 0.785695i 0.195090 0.980785i \(-0.437500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −2.36783 1.58213i −2.36783 1.58213i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(75\) 1.38704 1.38704i 1.38704 1.38704i
\(76\) −0.382683 0.923880i −0.382683 0.923880i
\(77\) 0 0
\(78\) −3.19929 0.636379i −3.19929 0.636379i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.00000 1.00000
\(81\) −4.26197 −4.26197
\(82\) 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
\(88\) 0.149316 + 0.750661i 0.149316 + 0.750661i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 2.36783 1.58213i 2.36783 1.58213i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 1.00000
\(96\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(97\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(98\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(99\) 1.54120 + 1.54120i 1.54120 + 1.54120i
\(100\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(101\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(102\) 0 0
\(103\) 0.390181i 0.390181i −0.980785 0.195090i \(-0.937500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(104\) 1.63099 0.324423i 1.63099 0.324423i
\(105\) 0 0
\(106\) 1.08979 + 1.63099i 1.08979 + 1.63099i
\(107\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(108\) 3.34861 1.38704i 3.34861 1.38704i
\(109\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(110\) −0.750661 0.149316i −0.750661 0.149316i
\(111\) −0.765367 −0.765367
\(112\) 0 0
\(113\) −1.96157 −1.96157 −0.980785 0.195090i \(-0.937500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(114\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(115\) 0 0
\(116\) 0 0
\(117\) 3.34861 3.34861i 3.34861 3.34861i
\(118\) 0 0
\(119\) 0 0
\(120\) −1.08979 + 1.63099i −1.08979 + 1.63099i
\(121\) 0.414214i 0.414214i
\(122\) −0.636379 + 0.425215i −0.636379 + 0.425215i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.707107 0.707107i −0.707107 0.707107i
\(126\) 0 0
\(127\) 1.11114 1.11114 0.555570 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(128\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(129\) 0 0
\(130\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(131\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(132\) −1.38704 0.574531i −1.38704 0.574531i
\(133\) 0 0
\(134\) −0.923880 + 0.617317i −0.923880 + 0.617317i
\(135\) 3.62451i 3.62451i
\(136\) 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 \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.27276 −1.27276
\(144\) −2.01367 + 2.01367i −2.01367 + 2.01367i
\(145\) 0 0
\(146\) 0 0
\(147\) −1.38704 1.38704i −1.38704 1.38704i
\(148\) 0.360480 0.149316i 0.360480 0.149316i
\(149\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(150\) −1.08979 1.63099i −1.08979 1.63099i
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.24830 + 3.01367i −1.24830 + 3.01367i
\(157\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) −3.84776 −3.84776
\(160\) 0.195090 0.980785i 0.195090 0.980785i
\(161\) 0 0
\(162\) −0.831470 + 4.18008i −0.831470 + 4.18008i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 1.06159 1.06159i 1.06159 1.06159i
\(166\) 0 0
\(167\) 1.11114i 1.11114i 0.831470 + 0.555570i \(0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(168\) 0 0
\(169\) 1.76537i 1.76537i
\(170\) 0 0
\(171\) −2.01367 + 2.01367i −2.01367 + 2.01367i
\(172\) 0 0
\(173\) 0.785695 + 0.785695i 0.785695 + 0.785695i 0.980785 0.195090i \(-0.0625000\pi\)
−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 \(0.250000\pi\)
\(180\) −1.08979 2.63099i −1.08979 2.63099i
\(181\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(182\) 0 0
\(183\) 1.50132i 1.50132i
\(184\) 0 0
\(185\) 0.390181i 0.390181i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0.195090 0.980785i 0.195090 0.980785i
\(191\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(192\) 0.750661 1.81225i 0.750661 1.81225i
\(193\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(194\) 0.324423 1.63099i 0.324423 1.63099i
\(195\) −2.30656 2.30656i −2.30656 2.30656i
\(196\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 1.81225 1.21091i 1.81225 1.21091i
\(199\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(200\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(201\) 2.17958i 2.17958i
\(202\) 0.425215 + 0.636379i 0.425215 + 0.636379i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −0.382683 0.0761205i −0.382683 0.0761205i
\(207\) 0 0
\(208\) 1.66294i 1.66294i
\(209\) 0.765367 0.765367
\(210\) 0 0
\(211\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(212\) 1.81225 0.750661i 1.81225 0.750661i
\(213\) 0 0
\(214\) −0.617317 0.923880i −0.617317 0.923880i
\(215\) 0 0
\(216\) −0.707107 3.55487i −0.707107 3.55487i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(221\) 0 0
\(222\) −0.149316 + 0.750661i −0.149316 + 0.750661i
\(223\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(224\) 0 0
\(225\) 2.84776 2.84776
\(226\) −0.382683 + 1.92388i −0.382683 + 1.92388i
\(227\) −1.17588 1.17588i −1.17588 1.17588i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(228\) 0.750661 1.81225i 0.750661 1.81225i
\(229\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −2.63099 3.93755i −2.63099 3.93755i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0.406255 + 0.0808091i 0.406255 + 0.0808091i
\(243\) −3.34861 3.34861i −3.34861 3.34861i
\(244\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(245\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(246\) 0 0
\(247\) 1.66294i 1.66294i
\(248\) 0 0
\(249\) 0 0
\(250\) −0.831470 + 0.555570i −0.831470 + 0.555570i
\(251\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.216773 1.08979i 0.216773 1.08979i
\(255\) 0 0
\(256\) 1.00000i 1.00000i
\(257\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.53636 + 0.636379i 1.53636 + 0.636379i
\(261\) 0 0
\(262\) −1.17588 + 0.785695i −1.17588 + 0.785695i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −0.834089 + 1.24830i −0.834089 + 1.24830i
\(265\) 1.96157i 1.96157i
\(266\) 0 0
\(267\) 0 0
\(268\) 0.425215 + 1.02656i 0.425215 + 1.02656i
\(269\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(270\) 3.55487 + 0.707107i 3.55487 + 0.707107i
\(271\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(272\) 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 \(0.750000\pi\)
\(278\) −1.02656 1.53636i −1.02656 1.53636i
\(279\) 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
\(285\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(286\) −0.248303 + 1.24830i −0.248303 + 1.24830i
\(287\) 0 0
\(288\) 1.58213 + 2.36783i 1.58213 + 2.36783i
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) 2.30656 + 2.30656i 2.30656 + 2.30656i
\(292\) 0 0
\(293\) 0.275899 0.275899i 0.275899 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(294\) −1.63099 + 1.08979i −1.63099 + 1.08979i
\(295\) 0 0
\(296\) −0.0761205 0.382683i −0.0761205 0.382683i
\(297\) 2.77408i 2.77408i
\(298\) −1.02656 1.53636i −1.02656 1.53636i
\(299\) 0 0
\(300\) −1.81225 + 0.750661i −1.81225 + 0.750661i
\(301\) 0 0
\(302\) 0 0
\(303\) −1.50132 −1.50132
\(304\) 1.00000i 1.00000i
\(305\) −0.765367 −0.765367
\(306\) 0 0
\(307\) 1.17588 + 1.17588i 1.17588 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(308\) 0 0
\(309\) 0.541196 0.541196i 0.541196 0.541196i
\(310\) 0 0
\(311\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(312\) 2.71223 + 1.81225i 2.71223 + 1.81225i
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.785695 + 0.785695i 0.785695 + 0.785695i 0.980785 0.195090i \(-0.0625000\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(318\) −0.750661 + 3.77383i −0.750661 + 3.77383i
\(319\) 0 0
\(320\) −0.923880 0.382683i −0.923880 0.382683i
\(321\) 2.17958 2.17958
\(322\) 0 0
\(323\) 0 0
\(324\) 3.93755 + 1.63099i 3.93755 + 1.63099i
\(325\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) −0.834089 1.24830i −0.834089 1.24830i
\(331\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) 0 0
\(333\) −0.785695 0.785695i −0.785695 0.785695i
\(334\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(335\) −1.11114 −1.11114
\(336\) 0 0
\(337\) 0.390181 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(338\) 1.73145 + 0.344406i 1.73145 + 0.344406i
\(339\) −2.72078 2.72078i −2.72078 2.72078i
\(340\) 0 0
\(341\) 0 0
\(342\) 1.58213 + 2.36783i 1.58213 + 2.36783i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.923880 0.617317i 0.923880 0.617317i
\(347\) 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
\(351\) 6.02734 6.02734
\(352\) 0.149316 0.750661i 0.149316 0.750661i
\(353\) 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 \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(360\) −2.79304 + 0.555570i −2.79304 + 0.555570i
\(361\) 1.00000i 1.00000i
\(362\) 0 0
\(363\) −0.574531 + 0.574531i −0.574531 + 0.574531i
\(364\) 0 0
\(365\) 0 0
\(366\) −1.47247 0.292893i −1.47247 0.292893i
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(371\) 0 0
\(372\) 0 0
\(373\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(374\) 0 0
\(375\) 1.96157i 1.96157i
\(376\) 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
\(381\) 1.54120 + 1.54120i 1.54120 + 1.54120i
\(382\) −0.275899 + 1.38704i −0.275899 + 1.38704i
\(383\) 0.390181 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(384\) −1.63099 1.08979i −1.63099 1.08979i
\(385\) 0 0
\(386\) −0.216773 + 1.08979i −0.216773 + 1.08979i
\(387\) 0 0
\(388\) −1.53636 0.636379i −1.53636 0.636379i
\(389\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(390\) −2.71223 + 1.81225i −2.71223 + 1.81225i
\(391\) 0 0
\(392\) 0.555570 0.831470i 0.555570 0.831470i
\(393\) 2.77408i 2.77408i
\(394\) 0 0
\(395\) 0 0
\(396\) −0.834089 2.01367i −0.834089 2.01367i
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 1.38704 + 0.275899i 1.38704 + 0.275899i
\(399\) 0 0
\(400\) 0.707107 0.707107i 0.707107 0.707107i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) −2.13770 0.425215i −2.13770 0.425215i
\(403\) 0 0
\(404\) 0.707107 0.292893i 0.707107 0.292893i
\(405\) −3.01367 + 3.01367i −3.01367 + 3.01367i
\(406\) 0 0
\(407\) 0.298631i 0.298631i
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.149316 + 0.360480i −0.149316 + 0.360480i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.63099 0.324423i −1.63099 0.324423i
\(417\) 3.62451 3.62451
\(418\) 0.149316 0.750661i 0.149316 0.750661i
\(419\) 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 1.92388i −0.382683 1.92388i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.02656 + 0.425215i −1.02656 + 0.425215i
\(429\) −1.76537 1.76537i −1.76537 1.76537i
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −3.62451 −3.62451
\(433\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0.636379 + 0.425215i 0.636379 + 0.425215i
\(441\) 2.84776i 2.84776i
\(442\) 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
\(446\) 0.324423 1.63099i 0.324423 1.63099i
\(447\) 3.62451 3.62451
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0.555570 2.79304i 0.555570 2.79304i
\(451\) 0 0
\(452\) 1.81225 + 0.750661i 1.81225 + 0.750661i
\(453\) 0 0
\(454\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(455\) 0 0
\(456\) −1.63099 1.08979i −1.63099 1.08979i
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 1.02656 + 1.53636i 1.02656 + 1.53636i
\(459\) 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 \(0.250000\pi\)
\(468\) −4.37517 + 1.81225i −4.37517 + 1.81225i
\(469\) 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
\(477\) −3.94996 3.94996i −3.94996 3.94996i
\(478\) 0 0
\(479\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(480\) 1.63099 1.08979i 1.63099 1.08979i
\(481\) 0.648847 0.648847
\(482\) 0 0
\(483\) 0 0
\(484\) 0.158513 0.382683i 0.158513 0.382683i
\(485\) 1.17588 1.17588i 1.17588 1.17588i
\(486\) −3.93755 + 2.63099i −3.93755 + 2.63099i
\(487\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(488\) 0.750661 0.149316i 0.750661 0.149316i
\(489\) 0 0
\(490\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(491\) 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
\(494\) −1.63099 0.324423i −1.63099 0.324423i
\(495\) 2.17958 2.17958
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(500\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(501\) −1.54120 + 1.54120i −1.54120 + 1.54120i
\(502\) 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
\(507\) −2.44863 + 2.44863i −2.44863 + 2.44863i
\(508\) −1.02656 0.425215i −1.02656 0.425215i
\(509\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(513\) −3.62451 −3.62451
\(514\) −0.216773 + 1.08979i −0.216773 + 1.08979i
\(515\) −0.275899 0.275899i −0.275899 0.275899i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.17958i 2.17958i
\(520\) 0.923880 1.38268i 0.923880 1.38268i
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −0.275899 + 0.275899i −0.275899 + 0.275899i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(524\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.06159 + 1.06159i 1.06159 + 1.06159i
\(529\) −1.00000 −1.00000
\(530\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.11114i 1.11114i
\(536\) 1.08979 0.216773i 1.08979 0.216773i
\(537\) 0 0
\(538\) 0 0
\(539\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(540\) 1.38704 3.34861i 1.38704 3.34861i
\(541\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(542\) 0.360480 1.81225i 0.360480 1.81225i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.275899 + 0.275899i 0.275899 + 0.275899i 0.831470 0.555570i \(-0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(548\) 0 0
\(549\) 1.54120 1.54120i 1.54120 1.54120i
\(550\) −0.636379 + 0.425215i −0.636379 + 0.425215i
\(551\) 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
\(563\) −0.275899 0.275899i −0.275899 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(564\) 0 0
\(565\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 1.63099 1.08979i 1.63099 1.08979i
\(571\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(572\) 1.17588 + 0.487064i 1.17588 + 0.487064i
\(573\) −1.96157 1.96157i −1.96157 1.96157i
\(574\) 0 0
\(575\) 0 0
\(576\) 2.63099 1.08979i 2.63099 1.08979i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0.195090 0.980785i 0.195090 0.980785i
\(579\) −1.54120 1.54120i −1.54120 1.54120i
\(580\) 0 0
\(581\) 0 0
\(582\) 2.71223 1.81225i 2.71223 1.81225i
\(583\) 1.50132i 1.50132i
\(584\) 0 0
\(585\) 4.73565i 4.73565i
\(586\) −0.216773 0.324423i −0.216773 0.324423i
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0.750661 + 1.81225i 0.750661 + 1.81225i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.390181 −0.390181
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 2.72078 + 0.541196i 2.72078 + 0.541196i
\(595\) 0 0
\(596\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(597\) −1.96157 + 1.96157i −1.96157 + 1.96157i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0.382683 + 1.92388i 0.382683 + 1.92388i
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 2.23747 2.23747i 2.23747 2.23747i
\(604\) 0 0
\(605\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(606\) −0.292893 + 1.47247i −0.292893 + 1.47247i
\(607\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(608\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(609\) 0 0
\(610\) −0.149316 + 0.750661i −0.149316 + 0.750661i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 1.38268 0.923880i 1.38268 0.923880i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) −0.425215 0.636379i −0.425215 0.636379i
\(619\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.750661 + 0.149316i 0.750661 + 0.149316i
\(623\) 0 0
\(624\) 2.30656 2.30656i 2.30656 2.30656i
\(625\) −1.00000 −1.00000
\(626\) 0 0
\(627\) 1.06159 + 1.06159i 1.06159 + 1.06159i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.923880 0.617317i 0.923880 0.617317i
\(635\) 0.785695 0.785695i 0.785695 0.785695i
\(636\) 3.55487 + 1.47247i 3.55487 + 1.47247i
\(637\) 1.17588 + 1.17588i 1.17588 + 1.17588i
\(638\) 0 0
\(639\) 0 0
\(640\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0.425215 2.13770i 0.425215 2.13770i
\(643\) 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 \(0.5\pi\)
\(648\) 2.36783 3.54370i 2.36783 3.54370i
\(649\) 0 0
\(650\) 0.923880 + 1.38268i 0.923880 + 1.38268i
\(651\) 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 \(0.250000\pi\)
\(660\) −1.38704 + 0.574531i −1.38704 + 0.574531i
\(661\) 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 + 0.617317i −0.923880 + 0.617317i
\(667\) 0 0
\(668\) 0.425215 1.02656i 0.425215 1.02656i
\(669\) 2.30656 + 2.30656i 2.30656 + 2.30656i
\(670\) −0.216773 + 1.08979i −0.216773 + 1.08979i
\(671\) −0.585786 −0.585786
\(672\) 0 0
\(673\) −0.390181 −0.390181 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(674\) 0.0761205 0.382683i 0.0761205 0.382683i
\(675\) 2.56292 + 2.56292i 2.56292 + 2.56292i
\(676\) 0.675577 1.63099i 0.675577 1.63099i
\(677\) −0.275899 + 0.275899i −0.275899 + 0.275899i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(678\) −3.19929 + 2.13770i −3.19929 + 2.13770i
\(679\) 0 0
\(680\) 0 0
\(681\) 3.26197i 3.26197i
\(682\) 0 0
\(683\) 0.275899 0.275899i 0.275899 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(684\) 2.63099 1.08979i 2.63099 1.08979i
\(685\) 0 0
\(686\) 0 0
\(687\) −3.62451 −3.62451
\(688\) 0 0
\(689\) 3.26197 3.26197
\(690\) 0 0
\(691\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(692\) −0.425215 1.02656i −0.425215 1.02656i
\(693\) 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 \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(702\) 1.17588 5.91153i 1.17588 5.91153i
\(703\) −0.390181 −0.390181
\(704\) −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
\(715\) −0.899976 + 0.899976i −0.899976 + 0.899976i
\(716\) 0 0
\(717\) 0 0
\(718\) 1.81225 + 0.360480i 1.81225 + 0.360480i
\(719\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(720\) 2.84776i 2.84776i
\(721\) 0 0
\(722\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0.451406 + 0.675577i 0.451406 + 0.675577i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 5.02734i 5.02734i
\(730\) 0 0
\(731\) 0 0
\(732\) −0.574531 + 1.38704i −0.574531 + 1.38704i
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) −1.96157 −1.96157
\(736\) 0 0
\(737\) −0.850430 −0.850430
\(738\) 0 0
\(739\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(740\) 0.149316 0.360480i 0.149316 0.360480i
\(741\) 2.30656 2.30656i 2.30656 2.30656i
\(742\) 0 0
\(743\) 1.66294i 1.66294i 0.555570 + 0.831470i \(0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(744\) 0 0
\(745\) 1.84776i 1.84776i
\(746\) −1.08979 1.63099i −1.08979 1.63099i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −1.92388 0.382683i −1.92388 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
\(760\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(761\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(762\) 1.81225 1.21091i 1.81225 1.21091i
\(763\) 0 0
\(764\) 1.30656 + 0.541196i 1.30656 + 0.541196i
\(765\) 0 0
\(766\) 0.0761205 0.382683i 0.0761205 0.382683i
\(767\) 0 0
\(768\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(769\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(770\) 0 0
\(771\) −1.54120 1.54120i −1.54120 1.54120i
\(772\) 1.02656 + 0.425215i 1.02656 + 0.425215i
\(773\) 1.17588 1.17588i 1.17588 1.17588i 0.195090 0.980785i \(-0.437500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.923880 + 1.38268i −0.923880 + 1.38268i
\(777\) 0 0
\(778\) −0.785695 1.17588i −0.785695 1.17588i
\(779\) 0 0
\(780\) 1.24830 + 3.01367i 1.24830 + 3.01367i
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.707107 0.707107i −0.707107 0.707107i
\(785\) 0 0
\(786\) −2.72078 0.541196i −2.72078 0.541196i
\(787\) 0.275899 + 0.275899i 0.275899 + 0.275899i 0.831470 0.555570i \(-0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −2.13770 + 0.425215i −2.13770 + 0.425215i
\(793\) 1.27276i 1.27276i
\(794\) 0 0
\(795\) −2.72078 + 2.72078i −2.72078 + 2.72078i
\(796\) 0.541196 1.30656i 0.541196 1.30656i
\(797\) −0.275899 0.275899i −0.275899 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.555570 0.831470i −0.555570 0.831470i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.834089 + 2.01367i −0.834089 + 2.01367i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.149316 0.750661i −0.149316 0.750661i
\(809\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(810\) 2.36783 + 3.54370i 2.36783 + 3.54370i
\(811\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) 2.56292 + 2.56292i 2.56292 + 2.56292i
\(814\) 0.292893 + 0.0582601i 0.292893 + 0.0582601i
\(815\) 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 \(0.5\pi\)
\(824\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(825\) 1.50132i 1.50132i
\(826\) 0 0
\(827\) 0.275899 0.275899i 0.275899 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
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
\(832\) −0.636379 + 1.53636i −0.636379 + 1.53636i
\(833\) 0 0
\(834\) 0.707107 3.55487i 0.707107 3.55487i
\(835\) 0.785695 + 0.785695i 0.785695 + 0.785695i
\(836\) −0.707107 0.292893i −0.707107 0.292893i
\(837\) 0 0
\(838\) 1.17588 0.785695i 1.17588 0.785695i
\(839\) 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
\(845\) 1.24830 + 1.24830i 1.24830 + 1.24830i
\(846\) 0 0
\(847\) 0 0
\(848\) −1.96157 −1.96157
\(849\) 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
\(855\) 2.84776i 2.84776i
\(856\) 0.216773 + 1.08979i 0.216773 + 1.08979i
\(857\) 1.11114i 1.11114i 0.831470 + 0.555570i \(0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(858\) −2.07585 + 1.38704i −2.07585 + 1.38704i
\(859\) 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
\(863\) 1.11114 1.11114 0.555570 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(864\) −0.707107 + 3.55487i −0.707107 + 3.55487i
\(865\) 1.11114 1.11114
\(866\) 0.382683 1.92388i 0.382683 1.92388i
\(867\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 1.84776i 1.84776i
\(872\) 0 0
\(873\) 4.73565i 4.73565i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.38704 1.38704i −1.38704 1.38704i −0.831470 0.555570i \(-0.812500\pi\)
−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 \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(882\) −2.79304 0.555570i −2.79304 0.555570i
\(883\) 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
\(887\) 1.96157i 1.96157i 0.195090 + 0.980785i \(0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(888\) 0.425215 0.636379i 0.425215 0.636379i
\(889\) 0 0
\(890\) 0 0
\(891\) −2.30656 + 2.30656i −2.30656 + 2.30656i
\(892\) −1.53636 0.636379i −1.53636 0.636379i
\(893\) 0 0
\(894\) 0.707107 3.55487i 0.707107 3.55487i
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −2.63099 1.08979i −2.63099 1.08979i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 1.08979 1.63099i 1.08979 1.63099i
\(905\) 0 0
\(906\) 0 0
\(907\) 1.17588 1.17588i 1.17588 1.17588i 0.195090 0.980785i \(-0.437500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(908\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(909\) −1.54120 1.54120i −1.54120 1.54120i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(913\) 0 0
\(914\) 0 0
\(915\) −1.06159 1.06159i −1.06159 1.06159i
\(916\) 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
\(921\) 3.26197i 3.26197i
\(922\) −1.17588 + 0.785695i −1.17588 + 0.785695i
\(923\) 0 0
\(924\) 0 0
\(925\) 0.275899 + 0.275899i 0.275899 + 0.275899i
\(926\) 0 0
\(927\) 1.11114 1.11114
\(928\) 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
\(933\) −1.06159 + 1.06159i −1.06159 + 1.06159i
\(934\) 0 0
\(935\) 0 0
\(936\) 0.923880 + 4.64466i 0.923880 + 4.64466i
\(937\) 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
\(950\) −0.555570 0.831470i −0.555570 0.831470i
\(951\) 2.17958i 2.17958i
\(952\) 0 0
\(953\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(954\) −4.64466 + 3.10346i −4.64466 + 3.10346i
\(955\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(956\) 0 0
\(957\) 0 0
\(958\) −0.149316 + 0.750661i −0.149316 + 0.750661i
\(959\) 0 0
\(960\) −0.750661 1.81225i −0.750661 1.81225i
\(961\) 1.00000 1.00000
\(962\) 0.126584 0.636379i 0.126584 0.636379i
\(963\) 2.23747 + 2.23747i 2.23747 + 2.23747i
\(964\) 0 0
\(965\) −0.785695 + 0.785695i −0.785695 + 0.785695i
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −0.344406 0.230125i −0.344406 0.230125i
\(969\) 0 0
\(970\) −0.923880 1.38268i −0.923880 1.38268i
\(971\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(972\) 1.81225 + 4.37517i 1.81225 + 4.37517i
\(973\) 0 0
\(974\) −1.92388 0.382683i −1.92388 0.382683i
\(975\) −3.26197 −3.26197
\(976\) 0.765367i 0.765367i
\(977\) −0.390181 −0.390181 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.923880 0.382683i 0.923880 0.382683i
\(981\) 0 0
\(982\) −1.11114 1.66294i −1.11114 1.66294i
\(983\) 0.390181i 0.390181i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −0.636379 + 1.53636i −0.636379 + 1.53636i
\(989\) 0 0
\(990\) 0.425215 2.13770i 0.425215 2.13770i
\(991\) 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 \(0.750000\pi\)
\(998\) 1.53636 1.02656i 1.53636 1.02656i
\(999\) 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.189.5 yes 16
5.4 even 2 inner 1520.1.bh.c.189.4 16
16.5 even 4 inner 1520.1.bh.c.949.5 yes 16
19.18 odd 2 inner 1520.1.bh.c.189.4 16
80.69 even 4 inner 1520.1.bh.c.949.4 yes 16
95.94 odd 2 CM 1520.1.bh.c.189.5 yes 16
304.37 odd 4 inner 1520.1.bh.c.949.4 yes 16
1520.949 odd 4 inner 1520.1.bh.c.949.5 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1520.1.bh.c.189.4 16 5.4 even 2 inner
1520.1.bh.c.189.4 16 19.18 odd 2 inner
1520.1.bh.c.189.5 yes 16 1.1 even 1 trivial
1520.1.bh.c.189.5 yes 16 95.94 odd 2 CM
1520.1.bh.c.949.4 yes 16 80.69 even 4 inner
1520.1.bh.c.949.4 yes 16 304.37 odd 4 inner
1520.1.bh.c.949.5 yes 16 16.5 even 4 inner
1520.1.bh.c.949.5 yes 16 1520.949 odd 4 inner