Properties

Label 3584.1.v.a
Level $3584$
Weight $1$
Character orbit 3584.v
Analytic conductor $1.789$
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] = [3584,1,Mod(321,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, 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("3584.321");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3584.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: \(1.78864900528\)
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 \( q - \zeta_{8}^{3} q^{7} - \zeta_{8} q^{9} + ( - \zeta_{8}^{3} - 1) q^{11} + ( - \zeta_{8}^{2} - 1) q^{23} - \zeta_{8}^{3} q^{25} + (\zeta_{8} - 1) q^{29} + (\zeta_{8}^{3} - 1) q^{37} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{43} - \zeta_{8}^{2} q^{49} + (\zeta_{8}^{2} + \zeta_{8}) q^{53} - q^{63} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{67} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{77} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{79} + \zeta_{8}^{2} q^{81} + (\zeta_{8} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{11} - 4 q^{23} - 4 q^{29} - 4 q^{37} - 4 q^{63} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{8}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
−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
1217.1 0 0 0 0 0 −0.707107 0.707107i 0 0.707107 0.707107i 0
2113.1 0 0 0 0 0 0.707107 0.707107i 0 −0.707107 0.707107i 0
3009.1 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 3584.1.v.a 4
4.b odd 2 1 3584.1.v.d 4
7.b odd 2 1 CM 3584.1.v.a 4
8.b even 2 1 3584.1.v.c 4
8.d odd 2 1 3584.1.v.b 4
16.e even 4 1 896.1.v.a 4
16.e even 4 1 1792.1.v.b 4
16.f odd 4 1 224.1.v.a 4
16.f odd 4 1 1792.1.v.a 4
28.d even 2 1 3584.1.v.d 4
32.g even 8 1 896.1.v.a 4
32.g even 8 1 1792.1.v.b 4
32.g even 8 1 inner 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
48.k even 4 1 2016.1.dp.b 4
56.e even 2 1 3584.1.v.b 4
56.h odd 2 1 3584.1.v.c 4
96.o even 8 1 2016.1.dp.b 4
112.j even 4 1 224.1.v.a 4
112.j even 4 1 1792.1.v.a 4
112.l odd 4 1 896.1.v.a 4
112.l odd 4 1 1792.1.v.b 4
112.u odd 12 2 1568.1.bl.a 8
112.v even 12 2 1568.1.bl.a 8
224.v odd 8 1 896.1.v.a 4
224.v odd 8 1 1792.1.v.b 4
224.v odd 8 1 inner 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
336.v odd 4 1 2016.1.dp.b 4
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 16.f odd 4 1
224.1.v.a 4 32.h odd 8 1
224.1.v.a 4 112.j even 4 1
224.1.v.a 4 224.x even 8 1
896.1.v.a 4 16.e even 4 1
896.1.v.a 4 32.g even 8 1
896.1.v.a 4 112.l odd 4 1
896.1.v.a 4 224.v odd 8 1
1568.1.bl.a 8 112.u odd 12 2
1568.1.bl.a 8 112.v even 12 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 16.f odd 4 1
1792.1.v.a 4 32.h odd 8 1
1792.1.v.a 4 112.j even 4 1
1792.1.v.a 4 224.x even 8 1
1792.1.v.b 4 16.e even 4 1
1792.1.v.b 4 32.g even 8 1
1792.1.v.b 4 112.l odd 4 1
1792.1.v.b 4 224.v odd 8 1
2016.1.dp.b 4 48.k even 4 1
2016.1.dp.b 4 96.o even 8 1
2016.1.dp.b 4 336.v odd 4 1
2016.1.dp.b 4 672.br odd 8 1
3584.1.v.a 4 1.a even 1 1 trivial
3584.1.v.a 4 7.b odd 2 1 CM
3584.1.v.a 4 32.g even 8 1 inner
3584.1.v.a 4 224.v odd 8 1 inner
3584.1.v.b 4 8.d odd 2 1
3584.1.v.b 4 32.h odd 8 1
3584.1.v.b 4 56.e even 2 1
3584.1.v.b 4 224.x even 8 1
3584.1.v.c 4 8.b even 2 1
3584.1.v.c 4 32.g even 8 1
3584.1.v.c 4 56.h odd 2 1
3584.1.v.c 4 224.v odd 8 1
3584.1.v.d 4 4.b odd 2 1
3584.1.v.d 4 28.d even 2 1
3584.1.v.d 4 32.h odd 8 1
3584.1.v.d 4 224.x even 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3584, [\chi])\):

\( T_{11}^{4} + 4T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 2T_{23} + 2 \) Copy content Toggle raw display

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
$11$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) 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
$29$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{2} + 4 T + 2 \) 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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