Properties

Label 162.4.a.a
Level $162$
Weight $4$
Character orbit 162.a
Self dual yes
Analytic conductor $9.558$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(9.55830942093\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 9 q^{5} - 31 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + 9 q^{5} - 31 q^{7} - 8 q^{8} - 18 q^{10} + 15 q^{11} - 37 q^{13} + 62 q^{14} + 16 q^{16} + 42 q^{17} - 28 q^{19} + 36 q^{20} - 30 q^{22} - 195 q^{23} - 44 q^{25} + 74 q^{26} - 124 q^{28} - 111 q^{29} - 205 q^{31} - 32 q^{32} - 84 q^{34} - 279 q^{35} - 166 q^{37} + 56 q^{38} - 72 q^{40} + 261 q^{41} - 43 q^{43} + 60 q^{44} + 390 q^{46} - 177 q^{47} + 618 q^{49} + 88 q^{50} - 148 q^{52} - 114 q^{53} + 135 q^{55} + 248 q^{56} + 222 q^{58} - 159 q^{59} + 191 q^{61} + 410 q^{62} + 64 q^{64} - 333 q^{65} - 421 q^{67} + 168 q^{68} + 558 q^{70} - 156 q^{71} + 182 q^{73} + 332 q^{74} - 112 q^{76} - 465 q^{77} + 1133 q^{79} + 144 q^{80} - 522 q^{82} + 1083 q^{83} + 378 q^{85} + 86 q^{86} - 120 q^{88} + 1050 q^{89} + 1147 q^{91} - 780 q^{92} + 354 q^{94} - 252 q^{95} - 901 q^{97} - 1236 q^{98}+O(q^{100}) \) 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
−2.00000 0 4.00000 9.00000 0 −31.0000 −8.00000 0 −18.0000
\(n\): 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 162.4.a.a 1
3.b odd 2 1 162.4.a.d 1
4.b odd 2 1 1296.4.a.g 1
9.c even 3 2 54.4.c.a 2
9.d odd 6 2 18.4.c.a 2
12.b even 2 1 1296.4.a.b 1
36.f odd 6 2 432.4.i.a 2
36.h even 6 2 144.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a 2 9.d odd 6 2
54.4.c.a 2 9.c even 3 2
144.4.i.a 2 36.h even 6 2
162.4.a.a 1 1.a even 1 1 trivial
162.4.a.d 1 3.b odd 2 1
432.4.i.a 2 36.f odd 6 2
1296.4.a.b 1 12.b even 2 1
1296.4.a.g 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 9 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 9 \) Copy content Toggle raw display
$7$ \( T + 31 \) Copy content Toggle raw display
$11$ \( T - 15 \) Copy content Toggle raw display
$13$ \( T + 37 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T + 28 \) Copy content Toggle raw display
$23$ \( T + 195 \) Copy content Toggle raw display
$29$ \( T + 111 \) Copy content Toggle raw display
$31$ \( T + 205 \) Copy content Toggle raw display
$37$ \( T + 166 \) Copy content Toggle raw display
$41$ \( T - 261 \) Copy content Toggle raw display
$43$ \( T + 43 \) Copy content Toggle raw display
$47$ \( T + 177 \) Copy content Toggle raw display
$53$ \( T + 114 \) Copy content Toggle raw display
$59$ \( T + 159 \) Copy content Toggle raw display
$61$ \( T - 191 \) Copy content Toggle raw display
$67$ \( T + 421 \) Copy content Toggle raw display
$71$ \( T + 156 \) Copy content Toggle raw display
$73$ \( T - 182 \) Copy content Toggle raw display
$79$ \( T - 1133 \) Copy content Toggle raw display
$83$ \( T - 1083 \) Copy content Toggle raw display
$89$ \( T - 1050 \) Copy content Toggle raw display
$97$ \( T + 901 \) Copy content Toggle raw display
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