Properties

Label 243.1.b.a
Level $243$
Weight $1$
Character orbit 243.b
Self dual yes
Analytic conductor $0.121$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 243.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.121272798070\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.243.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.243.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{4} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{4} - q^{7} - q^{13} + q^{16} - q^{19} + q^{25} - q^{28} - q^{31} - q^{37} - q^{43} - q^{52} + 2 q^{61} + q^{64} + 2 q^{67} + 2 q^{73} - q^{76} - q^{79} + q^{91} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
242.1
0
0 0 1.00000 0 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.1.b.a 1
3.b odd 2 1 CM 243.1.b.a 1
4.b odd 2 1 3888.1.e.b 1
9.c even 3 2 243.1.d.a 2
9.d odd 6 2 243.1.d.a 2
12.b even 2 1 3888.1.e.b 1
27.e even 9 6 729.1.f.a 6
27.f odd 18 6 729.1.f.a 6
36.f odd 6 2 3888.1.q.b 2
36.h even 6 2 3888.1.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.1.b.a 1 1.a even 1 1 trivial
243.1.b.a 1 3.b odd 2 1 CM
243.1.d.a 2 9.c even 3 2
243.1.d.a 2 9.d odd 6 2
729.1.f.a 6 27.e even 9 6
729.1.f.a 6 27.f odd 18 6
3888.1.e.b 1 4.b odd 2 1
3888.1.e.b 1 12.b even 2 1
3888.1.q.b 2 36.f odd 6 2
3888.1.q.b 2 36.h even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(243, [\chi])\).

Hecke characteristic polynomials

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