Properties

Label 161.2.a.a
Level $161$
Weight $2$
Character orbit 161.a
Self dual yes
Analytic conductor $1.286$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.28559147254\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 2q^{5} + q^{7} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} - q^{4} + 2q^{5} + q^{7} + 3q^{8} - 3q^{9} - 2q^{10} + 4q^{11} + 6q^{13} - q^{14} - q^{16} - 2q^{17} + 3q^{18} + 4q^{19} - 2q^{20} - 4q^{22} - q^{23} - q^{25} - 6q^{26} - q^{28} - 2q^{29} - 4q^{31} - 5q^{32} + 2q^{34} + 2q^{35} + 3q^{36} - 2q^{37} - 4q^{38} + 6q^{40} - 6q^{41} + 12q^{43} - 4q^{44} - 6q^{45} + q^{46} - 12q^{47} + q^{49} + q^{50} - 6q^{52} - 10q^{53} + 8q^{55} + 3q^{56} + 2q^{58} + 2q^{61} + 4q^{62} - 3q^{63} + 7q^{64} + 12q^{65} + 12q^{67} + 2q^{68} - 2q^{70} + 8q^{71} - 9q^{72} - 14q^{73} + 2q^{74} - 4q^{76} + 4q^{77} + 8q^{79} - 2q^{80} + 9q^{81} + 6q^{82} - 4q^{83} - 4q^{85} - 12q^{86} + 12q^{88} + 6q^{89} + 6q^{90} + 6q^{91} + q^{92} + 12q^{94} + 8q^{95} - 10q^{97} - q^{98} - 12q^{99} + O(q^{100}) \)

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
−1.00000 0 −1.00000 2.00000 0 1.00000 3.00000 −3.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(23\) \(1\)

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 161.2.a.a 1
3.b odd 2 1 1449.2.a.d 1
4.b odd 2 1 2576.2.a.j 1
5.b even 2 1 4025.2.a.e 1
7.b odd 2 1 1127.2.a.a 1
23.b odd 2 1 3703.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.a 1 1.a even 1 1 trivial
1127.2.a.a 1 7.b odd 2 1
1449.2.a.d 1 3.b odd 2 1
2576.2.a.j 1 4.b odd 2 1
3703.2.a.a 1 23.b odd 2 1
4025.2.a.e 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(161))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( -1 + T \)
$11$ \( -4 + T \)
$13$ \( -6 + T \)
$17$ \( 2 + T \)
$19$ \( -4 + T \)
$23$ \( 1 + T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( -12 + T \)
$47$ \( 12 + T \)
$53$ \( 10 + T \)
$59$ \( T \)
$61$ \( -2 + T \)
$67$ \( -12 + T \)
$71$ \( -8 + T \)
$73$ \( 14 + T \)
$79$ \( -8 + T \)
$83$ \( 4 + T \)
$89$ \( -6 + T \)
$97$ \( 10 + T \)
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