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Properties

Label 1089.1.k.a
Level $1089$
Weight $1$
Character orbit 1089.k
Analytic conductor $0.543$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,1,Mod(118,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.118");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.k (of order \(10\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.3993.1
Artin image: $C_5\times \SD_{16}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{4} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{4} - \beta_{7} q^{7} - \beta_1 q^{13} + \beta_{4} q^{16} - \beta_{3} q^{19} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{25} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_1) q^{28} - \beta_{5} q^{43} - \beta_{4} q^{49} + \beta_{3} q^{52} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{61}+ \cdots + 2 \beta_{6} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} - 2 q^{16} + 2 q^{25} + 2 q^{49} - 2 q^{64} - 4 q^{91} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
0.831254 + 1.14412i
−0.831254 − 1.14412i
0.831254 − 1.14412i
−0.831254 + 1.14412i
−1.34500 + 0.437016i
1.34500 − 0.437016i
−1.34500 − 0.437016i
1.34500 + 0.437016i
0 0 0.309017 − 0.951057i 0 0 −1.34500 − 0.437016i 0 0 0
118.2 0 0 0.309017 − 0.951057i 0 0 1.34500 + 0.437016i 0 0 0
766.1 0 0 0.309017 + 0.951057i 0 0 −1.34500 + 0.437016i 0 0 0
766.2 0 0 0.309017 + 0.951057i 0 0 1.34500 − 0.437016i 0 0 0
820.1 0 0 −0.809017 + 0.587785i 0 0 −0.831254 − 1.14412i 0 0 0
820.2 0 0 −0.809017 + 0.587785i 0 0 0.831254 + 1.14412i 0 0 0
838.1 0 0 −0.809017 − 0.587785i 0 0 −0.831254 + 1.14412i 0 0 0
838.2 0 0 −0.809017 − 0.587785i 0 0 0.831254 − 1.14412i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.k.a 8
3.b odd 2 1 CM 1089.1.k.a 8
11.b odd 2 1 inner 1089.1.k.a 8
11.c even 5 1 1089.1.c.a ✓ 2
11.c even 5 3 inner 1089.1.k.a 8
11.d odd 10 1 1089.1.c.a ✓ 2
11.d odd 10 3 inner 1089.1.k.a 8
33.d even 2 1 inner 1089.1.k.a 8
33.f even 10 1 1089.1.c.a ✓ 2
33.f even 10 3 inner 1089.1.k.a 8
33.h odd 10 1 1089.1.c.a ✓ 2
33.h odd 10 3 inner 1089.1.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.c.a ✓ 2 11.c even 5 1
1089.1.c.a ✓ 2 11.d odd 10 1
1089.1.c.a ✓ 2 33.f even 10 1
1089.1.c.a ✓ 2 33.h odd 10 1
1089.1.k.a 8 1.a even 1 1 trivial
1089.1.k.a 8 3.b odd 2 1 CM
1089.1.k.a 8 11.b odd 2 1 inner
1089.1.k.a 8 11.c even 5 3 inner
1089.1.k.a 8 11.d odd 10 3 inner
1089.1.k.a 8 33.d even 2 1 inner
1089.1.k.a 8 33.f even 10 3 inner
1089.1.k.a 8 33.h odd 10 3 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1089, [\chi])\).

Hecke characteristic polynomials

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