Properties

Label 160.4.a.b
Level $160$
Weight $4$
Character orbit 160.a
Self dual yes
Analytic conductor $9.440$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.44030560092\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} - 5 q^{5} - 6 q^{7} - 23 q^{9} + O(q^{10}) \) \( q + 2 q^{3} - 5 q^{5} - 6 q^{7} - 23 q^{9} - 60 q^{11} + 50 q^{13} - 10 q^{15} - 30 q^{17} - 40 q^{19} - 12 q^{21} - 178 q^{23} + 25 q^{25} - 100 q^{27} + 166 q^{29} - 20 q^{31} - 120 q^{33} + 30 q^{35} + 10 q^{37} + 100 q^{39} - 250 q^{41} - 142 q^{43} + 115 q^{45} - 214 q^{47} - 307 q^{49} - 60 q^{51} + 490 q^{53} + 300 q^{55} - 80 q^{57} + 800 q^{59} + 250 q^{61} + 138 q^{63} - 250 q^{65} + 774 q^{67} - 356 q^{69} - 100 q^{71} - 230 q^{73} + 50 q^{75} + 360 q^{77} + 1320 q^{79} + 421 q^{81} - 982 q^{83} + 150 q^{85} + 332 q^{87} + 874 q^{89} - 300 q^{91} - 40 q^{93} + 200 q^{95} - 310 q^{97} + 1380 q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 2.00000 0 −5.00000 0 −6.00000 0 −23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.a.b yes 1
3.b odd 2 1 1440.4.a.n 1
4.b odd 2 1 160.4.a.a 1
5.b even 2 1 800.4.a.d 1
5.c odd 4 2 800.4.c.e 2
8.b even 2 1 320.4.a.f 1
8.d odd 2 1 320.4.a.i 1
12.b even 2 1 1440.4.a.o 1
16.e even 4 2 1280.4.d.k 2
16.f odd 4 2 1280.4.d.f 2
20.d odd 2 1 800.4.a.h 1
20.e even 4 2 800.4.c.f 2
40.e odd 2 1 1600.4.a.r 1
40.f even 2 1 1600.4.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 4.b odd 2 1
160.4.a.b yes 1 1.a even 1 1 trivial
320.4.a.f 1 8.b even 2 1
320.4.a.i 1 8.d odd 2 1
800.4.a.d 1 5.b even 2 1
800.4.a.h 1 20.d odd 2 1
800.4.c.e 2 5.c odd 4 2
800.4.c.f 2 20.e even 4 2
1280.4.d.f 2 16.f odd 4 2
1280.4.d.k 2 16.e even 4 2
1440.4.a.n 1 3.b odd 2 1
1440.4.a.o 1 12.b even 2 1
1600.4.a.r 1 40.e odd 2 1
1600.4.a.bj 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(160))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( 5 + T \)
$7$ \( 6 + T \)
$11$ \( 60 + T \)
$13$ \( -50 + T \)
$17$ \( 30 + T \)
$19$ \( 40 + T \)
$23$ \( 178 + T \)
$29$ \( -166 + T \)
$31$ \( 20 + T \)
$37$ \( -10 + T \)
$41$ \( 250 + T \)
$43$ \( 142 + T \)
$47$ \( 214 + T \)
$53$ \( -490 + T \)
$59$ \( -800 + T \)
$61$ \( -250 + T \)
$67$ \( -774 + T \)
$71$ \( 100 + T \)
$73$ \( 230 + T \)
$79$ \( -1320 + T \)
$83$ \( 982 + T \)
$89$ \( -874 + T \)
$97$ \( 310 + T \)
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