Properties

Label 243.1.d.a
Level $243$
Weight $1$
Character orbit 243.d
Analytic conductor $0.121$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 243.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.121272798070\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.243.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.177147.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{4} + \zeta_{6} q^{7} - \zeta_{6}^{2} q^{13} - \zeta_{6} q^{16} - q^{19} - \zeta_{6} q^{25} - q^{28} - \zeta_{6}^{2} q^{31} - q^{37} + \zeta_{6} q^{43} + \zeta_{6} q^{52} - \zeta_{6} q^{61} + q^{64} + \zeta_{6}^{2} q^{67} + q^{73} - \zeta_{6}^{2} q^{76} + \zeta_{6} q^{79} + q^{91} + \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{4} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{4} + q^{7} + q^{13} - q^{16} - 2 q^{19} - q^{25} - 2 q^{28} + q^{31} - 2 q^{37} + q^{43} + q^{52} - 2 q^{61} + 2 q^{64} - 2 q^{67} + 4 q^{73} + q^{76} + q^{79} + 2 q^{91} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −0.500000 0.866025i 0 0 0.500000 0.866025i 0 0 0
161.1 0 0 −0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 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 \(\Q(\sqrt{-3}) \)
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

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