Properties

Label 896.1.v.a
Level $896$
Weight $1$
Character orbit 896.v
Analytic conductor $0.447$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -7
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(209,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.209");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 896.v (of order \(8\), 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: \(0.447162251319\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.5156108238848.3

$q$-expansion

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

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

\(f(q)\) \(=\) \( q - \zeta_{8}^{3} q^{7} - \zeta_{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{3} q^{7} - \zeta_{8} q^{9} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{11} + (\zeta_{8}^{2} + 1) q^{23} - \zeta_{8}^{3} q^{25} + (\zeta_{8}^{3} + \zeta_{8}^{2}) q^{29} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{37} + (\zeta_{8}^{3} + 1) q^{43} - \zeta_{8}^{2} q^{49} + ( - \zeta_{8}^{3} - 1) q^{53} - q^{63} + (\zeta_{8} - 1) q^{67} + ( - \zeta_{8} - 1) q^{77} + (\zeta_{8}^{3} + \zeta_{8}) q^{79} + \zeta_{8}^{2} q^{81} + (\zeta_{8}^{3} + \zeta_{8}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{23} + 4 q^{43} - 4 q^{53} - 4 q^{63} - 4 q^{67} - 4 q^{77}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{8}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 0 0 −0.707107 + 0.707107i 0 0.707107 + 0.707107i 0
433.1 0 0 0 0 0 −0.707107 0.707107i 0 0.707107 0.707107i 0
657.1 0 0 0 0 0 0.707107 0.707107i 0 −0.707107 0.707107i 0
881.1 0 0 0 0 0 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
32.g even 8 1 inner
224.v odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.1.v.a 4
4.b odd 2 1 224.1.v.a 4
7.b odd 2 1 CM 896.1.v.a 4
8.b even 2 1 1792.1.v.b 4
8.d odd 2 1 1792.1.v.a 4
12.b even 2 1 2016.1.dp.b 4
16.e even 4 1 3584.1.v.a 4
16.e even 4 1 3584.1.v.c 4
16.f odd 4 1 3584.1.v.b 4
16.f odd 4 1 3584.1.v.d 4
28.d even 2 1 224.1.v.a 4
28.f even 6 2 1568.1.bl.a 8
28.g odd 6 2 1568.1.bl.a 8
32.g even 8 1 inner 896.1.v.a 4
32.g even 8 1 1792.1.v.b 4
32.g even 8 1 3584.1.v.a 4
32.g even 8 1 3584.1.v.c 4
32.h odd 8 1 224.1.v.a 4
32.h odd 8 1 1792.1.v.a 4
32.h odd 8 1 3584.1.v.b 4
32.h odd 8 1 3584.1.v.d 4
56.e even 2 1 1792.1.v.a 4
56.h odd 2 1 1792.1.v.b 4
84.h odd 2 1 2016.1.dp.b 4
96.o even 8 1 2016.1.dp.b 4
112.j even 4 1 3584.1.v.b 4
112.j even 4 1 3584.1.v.d 4
112.l odd 4 1 3584.1.v.a 4
112.l odd 4 1 3584.1.v.c 4
224.v odd 8 1 inner 896.1.v.a 4
224.v odd 8 1 1792.1.v.b 4
224.v odd 8 1 3584.1.v.a 4
224.v odd 8 1 3584.1.v.c 4
224.x even 8 1 224.1.v.a 4
224.x even 8 1 1792.1.v.a 4
224.x even 8 1 3584.1.v.b 4
224.x even 8 1 3584.1.v.d 4
224.be even 24 2 1568.1.bl.a 8
224.bf odd 24 2 1568.1.bl.a 8
672.br odd 8 1 2016.1.dp.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.v.a 4 4.b odd 2 1
224.1.v.a 4 28.d even 2 1
224.1.v.a 4 32.h odd 8 1
224.1.v.a 4 224.x even 8 1
896.1.v.a 4 1.a even 1 1 trivial
896.1.v.a 4 7.b odd 2 1 CM
896.1.v.a 4 32.g even 8 1 inner
896.1.v.a 4 224.v odd 8 1 inner
1568.1.bl.a 8 28.f even 6 2
1568.1.bl.a 8 28.g odd 6 2
1568.1.bl.a 8 224.be even 24 2
1568.1.bl.a 8 224.bf odd 24 2
1792.1.v.a 4 8.d odd 2 1
1792.1.v.a 4 32.h odd 8 1
1792.1.v.a 4 56.e even 2 1
1792.1.v.a 4 224.x even 8 1
1792.1.v.b 4 8.b even 2 1
1792.1.v.b 4 32.g even 8 1
1792.1.v.b 4 56.h odd 2 1
1792.1.v.b 4 224.v odd 8 1
2016.1.dp.b 4 12.b even 2 1
2016.1.dp.b 4 84.h odd 2 1
2016.1.dp.b 4 96.o even 8 1
2016.1.dp.b 4 672.br odd 8 1
3584.1.v.a 4 16.e even 4 1
3584.1.v.a 4 32.g even 8 1
3584.1.v.a 4 112.l odd 4 1
3584.1.v.a 4 224.v odd 8 1
3584.1.v.b 4 16.f odd 4 1
3584.1.v.b 4 32.h odd 8 1
3584.1.v.b 4 112.j even 4 1
3584.1.v.b 4 224.x even 8 1
3584.1.v.c 4 16.e even 4 1
3584.1.v.c 4 32.g even 8 1
3584.1.v.c 4 112.l odd 4 1
3584.1.v.c 4 224.v odd 8 1
3584.1.v.d 4 16.f odd 4 1
3584.1.v.d 4 32.h odd 8 1
3584.1.v.d 4 112.j even 4 1
3584.1.v.d 4 224.x even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(896, [\chi])\).

Hecke characteristic polynomials

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