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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Newspace parameters

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

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
Defining polynomial: \(x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16\)
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: $C_5\times {\rm SD}_{16}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

$q$-expansion

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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{4} -\beta_{7} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{4} -\beta_{7} q^{7} -\beta_{1} q^{13} + \beta_{4} q^{16} -\beta_{3} q^{19} + ( 1 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{25} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{28} -\beta_{5} q^{43} -\beta_{4} q^{49} + \beta_{3} q^{52} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{61} -\beta_{6} q^{64} + \beta_{7} q^{73} + \beta_{5} q^{76} + \beta_{1} q^{79} + ( -2 + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{91} + 2 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{4} + O(q^{10}) \) \( 8q - 2q^{4} - 2q^{16} + 2q^{25} + 2q^{49} - 2q^{64} - 4q^{91} + 4q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/4\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(4 \beta_{4}\)
\(\nu^{5}\)\(=\)\(4 \beta_{5}\)
\(\nu^{6}\)\(=\)\(8 \beta_{6}\)
\(\nu^{7}\)\(=\)\(8 \beta_{7}\)

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.

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 838.2
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

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