Properties

Label 896.1.h.b
Level $896$
Weight $1$
Character orbit 896.h
Self dual yes
Analytic conductor $0.447$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -56
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,1,Mod(321,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.321");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 896.h (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.1568.1
Artin image: $D_8$
Artin field: Galois closure of 8.0.1258815488.3

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - \beta q^{5} + q^{7} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - \beta q^{5} + q^{7} + q^{9} + \beta q^{13} - 2 q^{15} - \beta q^{19} + \beta q^{21} + q^{25} - \beta q^{35} + 2 q^{39} - \beta q^{45} + q^{49} - 2 q^{57} - \beta q^{59} - \beta q^{61} + q^{63} - 2 q^{65} - 2 q^{71} + \beta q^{75} - 2 q^{79} - q^{81} + \beta q^{83} + \beta q^{91} + 2 q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} + 2 q^{9} - 4 q^{15} + 2 q^{25} + 4 q^{39} + 2 q^{49} - 4 q^{57} + 2 q^{63} - 4 q^{65} - 4 q^{71} - 4 q^{79} - 2 q^{81} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
−1.41421
1.41421
0 −1.41421 0 1.41421 0 1.00000 0 1.00000 0
321.2 0 1.41421 0 −1.41421 0 1.00000 0 1.00000 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
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.h.b yes 2
4.b odd 2 1 896.1.h.a 2
7.b odd 2 1 inner 896.1.h.b yes 2
8.b even 2 1 inner 896.1.h.b yes 2
8.d odd 2 1 896.1.h.a 2
16.e even 4 2 1792.1.c.c 2
16.f odd 4 2 1792.1.c.d 2
28.d even 2 1 896.1.h.a 2
56.e even 2 1 896.1.h.a 2
56.h odd 2 1 CM 896.1.h.b yes 2
112.j even 4 2 1792.1.c.d 2
112.l odd 4 2 1792.1.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.1.h.a 2 4.b odd 2 1
896.1.h.a 2 8.d odd 2 1
896.1.h.a 2 28.d even 2 1
896.1.h.a 2 56.e even 2 1
896.1.h.b yes 2 1.a even 1 1 trivial
896.1.h.b yes 2 7.b odd 2 1 inner
896.1.h.b yes 2 8.b even 2 1 inner
896.1.h.b yes 2 56.h odd 2 1 CM
1792.1.c.c 2 16.e even 4 2
1792.1.c.c 2 112.l odd 4 2
1792.1.c.d 2 16.f odd 4 2
1792.1.c.d 2 112.j even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{71} + 2 \) acting on \(S_{1}^{\mathrm{new}}(896, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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