Properties

Label 162.2.a.d
Level $162$
Weight $2$
Character orbit 162.a
Self dual yes
Analytic conductor $1.294$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.29357651274\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 3 q^{5} - 4 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + 3 q^{5} - 4 q^{7} + q^{8} + 3 q^{10} - q^{13} - 4 q^{14} + q^{16} + 3 q^{17} - 4 q^{19} + 3 q^{20} + 4 q^{25} - q^{26} - 4 q^{28} - 9 q^{29} - 4 q^{31} + q^{32} + 3 q^{34} - 12 q^{35} - q^{37} - 4 q^{38} + 3 q^{40} - 6 q^{41} + 8 q^{43} + 12 q^{47} + 9 q^{49} + 4 q^{50} - q^{52} + 6 q^{53} - 4 q^{56} - 9 q^{58} - q^{61} - 4 q^{62} + q^{64} - 3 q^{65} - 4 q^{67} + 3 q^{68} - 12 q^{70} + 12 q^{71} + 11 q^{73} - q^{74} - 4 q^{76} - 16 q^{79} + 3 q^{80} - 6 q^{82} + 12 q^{83} + 9 q^{85} + 8 q^{86} + 3 q^{89} + 4 q^{91} + 12 q^{94} - 12 q^{95} + 2 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 3.00000 0 −4.00000 1.00000 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

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 162.2.a.d yes 1
3.b odd 2 1 162.2.a.a 1
4.b odd 2 1 1296.2.a.l 1
5.b even 2 1 4050.2.a.r 1
5.c odd 4 2 4050.2.c.n 2
7.b odd 2 1 7938.2.a.s 1
8.b even 2 1 5184.2.a.c 1
8.d odd 2 1 5184.2.a.h 1
9.c even 3 2 162.2.c.a 2
9.d odd 6 2 162.2.c.d 2
12.b even 2 1 1296.2.a.c 1
15.d odd 2 1 4050.2.a.bh 1
15.e even 4 2 4050.2.c.g 2
21.c even 2 1 7938.2.a.n 1
24.f even 2 1 5184.2.a.bd 1
24.h odd 2 1 5184.2.a.y 1
36.f odd 6 2 1296.2.i.b 2
36.h even 6 2 1296.2.i.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 3.b odd 2 1
162.2.a.d yes 1 1.a even 1 1 trivial
162.2.c.a 2 9.c even 3 2
162.2.c.d 2 9.d odd 6 2
1296.2.a.c 1 12.b even 2 1
1296.2.a.l 1 4.b odd 2 1
1296.2.i.b 2 36.f odd 6 2
1296.2.i.n 2 36.h even 6 2
4050.2.a.r 1 5.b even 2 1
4050.2.a.bh 1 15.d odd 2 1
4050.2.c.g 2 15.e even 4 2
4050.2.c.n 2 5.c odd 4 2
5184.2.a.c 1 8.b even 2 1
5184.2.a.h 1 8.d odd 2 1
5184.2.a.y 1 24.h odd 2 1
5184.2.a.bd 1 24.f even 2 1
7938.2.a.n 1 21.c even 2 1
7938.2.a.s 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(162))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 11 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 3 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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