Properties

Label 1089.1.c.a
Level $1089$
Weight $1$
Character orbit 1089.c
Analytic conductor $0.543$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1089.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.543481798757\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
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: $SD_{16}$
Artin field: Galois closure of 8.2.14206147659.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
604.1
1.41421i
1.41421i
0 0 1.00000 0 0 1.41421i 0 0 0
604.2 0 0 1.00000 0 0 1.41421i 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 \(\Q(\sqrt{-3}) \)
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.c.a 2
3.b odd 2 1 CM 1089.1.c.a 2
11.b odd 2 1 inner 1089.1.c.a 2
11.c even 5 4 1089.1.k.a 8
11.d odd 10 4 1089.1.k.a 8
33.d even 2 1 inner 1089.1.c.a 2
33.f even 10 4 1089.1.k.a 8
33.h odd 10 4 1089.1.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.c.a 2 1.a even 1 1 trivial
1089.1.c.a 2 3.b odd 2 1 CM
1089.1.c.a 2 11.b odd 2 1 inner
1089.1.c.a 2 33.d even 2 1 inner
1089.1.k.a 8 11.c even 5 4
1089.1.k.a 8 11.d odd 10 4
1089.1.k.a 8 33.f even 10 4
1089.1.k.a 8 33.h odd 10 4

Hecke kernels

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

Hecke characteristic polynomials

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