Properties

Label 729.1.f.a
Level $729$
Weight $1$
Character orbit 729.f
Analytic conductor $0.364$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -3
Inner twists $12$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 729.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.363818394209\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 243)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.243.1
Artin image: $C_9\times S_3$
Artin field: Galois closure of 18.0.2954312706550833698643.2

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{18} q^{4} - \zeta_{18}^{8} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18} q^{4} - \zeta_{18}^{8} q^{7} - \zeta_{18}^{4} q^{13} + \zeta_{18}^{2} q^{16} - \zeta_{18}^{6} q^{19} - \zeta_{18}^{5} q^{25} - q^{28} + \zeta_{18} q^{31} + \zeta_{18}^{3} q^{37} - \zeta_{18}^{2} q^{43} + \zeta_{18}^{5} q^{52} + \zeta_{18}^{8} q^{61} - \zeta_{18}^{3} q^{64} + \zeta_{18}^{4} q^{67} + \zeta_{18}^{6} q^{73} + \zeta_{18}^{7} q^{76} + \zeta_{18}^{5} q^{79} - \zeta_{18}^{3} q^{91} - \zeta_{18}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{19} - 6 q^{28} + 3 q^{37} - 3 q^{64} - 6 q^{73} - 3 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{18}\)

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
0.939693 + 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
0 0 −0.939693 0.342020i 0 0 0.939693 0.342020i 0 0 0
161.1 0 0 0.766044 + 0.642788i 0 0 −0.766044 + 0.642788i 0 0 0
323.1 0 0 0.173648 + 0.984808i 0 0 −0.173648 + 0.984808i 0 0 0
404.1 0 0 0.173648 0.984808i 0 0 −0.173648 0.984808i 0 0 0
566.1 0 0 0.766044 0.642788i 0 0 −0.766044 0.642788i 0 0 0
647.1 0 0 −0.939693 + 0.342020i 0 0 0.939693 + 0.342020i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 647.1
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 2 inner
9.d odd 6 2 inner
27.e even 9 3 inner
27.f odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.1.f.a 6
3.b odd 2 1 CM 729.1.f.a 6
9.c even 3 2 inner 729.1.f.a 6
9.d odd 6 2 inner 729.1.f.a 6
27.e even 9 1 243.1.b.a 1
27.e even 9 2 243.1.d.a 2
27.e even 9 3 inner 729.1.f.a 6
27.f odd 18 1 243.1.b.a 1
27.f odd 18 2 243.1.d.a 2
27.f odd 18 3 inner 729.1.f.a 6
108.j odd 18 1 3888.1.e.b 1
108.j odd 18 2 3888.1.q.b 2
108.l even 18 1 3888.1.e.b 1
108.l even 18 2 3888.1.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.1.b.a 1 27.e even 9 1
243.1.b.a 1 27.f odd 18 1
243.1.d.a 2 27.e even 9 2
243.1.d.a 2 27.f odd 18 2
729.1.f.a 6 1.a even 1 1 trivial
729.1.f.a 6 3.b odd 2 1 CM
729.1.f.a 6 9.c even 3 2 inner
729.1.f.a 6 9.d odd 6 2 inner
729.1.f.a 6 27.e even 9 3 inner
729.1.f.a 6 27.f odd 18 3 inner
3888.1.e.b 1 108.j odd 18 1
3888.1.e.b 1 108.l even 18 1
3888.1.q.b 2 108.j odd 18 2
3888.1.q.b 2 108.l even 18 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(729, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 8T^{3} + 64 \) Copy content Toggle raw display
$67$ \( T^{6} + 8T^{3} + 64 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$79$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
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