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Properties

Label 3584.2.a.h
Level $3584$
Weight $2$
Character orbit 3584.a
Self dual yes
Analytic conductor $28.618$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3584,2,Mod(1,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3584.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.a (trivial)

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: \(28.6183840844\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.12836864.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 22x^{2} - 8x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{5} - 1) q^{5} + q^{7} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{5} - 1) q^{5} + q^{7} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{5} + \beta_{4} + \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{5} + 6 q^{7} + 2 q^{9} - 4 q^{11} - 12 q^{13} - 8 q^{15} + 10 q^{25} - 16 q^{29} + 8 q^{31} - 4 q^{33} - 4 q^{35} - 16 q^{37} + 4 q^{41} - 12 q^{43} - 20 q^{45} + 8 q^{47} + 6 q^{49} - 8 q^{51} - 8 q^{53} - 16 q^{55} - 12 q^{57} - 44 q^{61} + 2 q^{63} - 20 q^{67} - 24 q^{69} - 8 q^{71} - 8 q^{73} + 40 q^{75} - 4 q^{77} - 24 q^{79} - 14 q^{81} + 16 q^{83} - 40 q^{85} + 8 q^{87} + 8 q^{89} - 12 q^{91} - 16 q^{93} + 16 q^{95} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 10x^{4} + 22x^{2} - 8x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 6\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 8\nu^{3} - 2\nu^{2} + 10\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 2\nu^{4} + 10\nu^{3} + 16\nu^{2} - 16\nu - 10 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 8\beta_{4} + 7\beta_{3} + \beta_{2} - 5\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{5} + 10\beta_{4} + 12\beta_{3} + 8\beta_{2} + 28\beta _1 + 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.36086
−2.14955
−0.284923
0.818115
1.23157
2.74564
0 −2.36086 0 0.711757 0 1.00000 0 2.57364 0
1.2 0 −2.14955 0 −2.09671 0 1.00000 0 1.62055 0
1.3 0 −0.284923 0 1.19247 0 1.00000 0 −2.91882 0
1.4 0 0.818115 0 3.08378 0 1.00000 0 −2.33069 0
1.5 0 1.23157 0 −3.90423 0 1.00000 0 −1.48325 0
1.6 0 2.74564 0 −2.98707 0 1.00000 0 4.53856 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.a.h yes 6
4.b odd 2 1 3584.2.a.g ✓ 6
8.b even 2 1 3584.2.a.j yes 6
8.d odd 2 1 3584.2.a.i yes 6
16.e even 4 2 3584.2.b.j 12
16.f odd 4 2 3584.2.b.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.g ✓ 6 4.b odd 2 1
3584.2.a.h yes 6 1.a even 1 1 trivial
3584.2.a.i yes 6 8.d odd 2 1
3584.2.a.j yes 6 8.b even 2 1
3584.2.b.j 12 16.e even 4 2
3584.2.b.l 12 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3584))\):

\( T_{3}^{6} - 10T_{3}^{4} + 22T_{3}^{2} - 8T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{6} + 4T_{5}^{5} - 12T_{5}^{4} - 48T_{5}^{3} + 30T_{5}^{2} + 96T_{5} - 64 \) Copy content Toggle raw display
\( T_{23}^{6} - 68T_{23}^{4} - 64T_{23}^{3} + 772T_{23}^{2} - 128T_{23} - 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 10 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{6} + 4 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} + 12 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$17$ \( T^{6} - 56 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$19$ \( T^{6} - 42 T^{4} + \cdots - 164 \) Copy content Toggle raw display
$23$ \( T^{6} - 68 T^{4} + \cdots - 1024 \) Copy content Toggle raw display
$29$ \( T^{6} + 16 T^{5} + \cdots - 4096 \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} + \cdots + 2048 \) Copy content Toggle raw display
$37$ \( T^{6} + 16 T^{5} + \cdots + 2048 \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} + \cdots - 3136 \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{5} + \cdots + 6592 \) Copy content Toggle raw display
$47$ \( T^{6} - 8 T^{5} + \cdots + 14336 \) Copy content Toggle raw display
$53$ \( T^{6} + 8 T^{5} + \cdots - 2048 \) Copy content Toggle raw display
$59$ \( T^{6} - 90 T^{4} + \cdots + 2428 \) Copy content Toggle raw display
$61$ \( T^{6} + 44 T^{5} + \cdots - 744512 \) Copy content Toggle raw display
$67$ \( T^{6} + 20 T^{5} + \cdots - 73984 \) Copy content Toggle raw display
$71$ \( T^{6} + 8 T^{5} + \cdots - 203776 \) Copy content Toggle raw display
$73$ \( T^{6} + 8 T^{5} + \cdots + 512 \) Copy content Toggle raw display
$79$ \( T^{6} + 24 T^{5} + \cdots - 171008 \) Copy content Toggle raw display
$83$ \( T^{6} - 16 T^{5} + \cdots - 4228 \) Copy content Toggle raw display
$89$ \( T^{6} - 8 T^{5} + \cdots - 52736 \) Copy content Toggle raw display
$97$ \( T^{6} - 520 T^{4} + \cdots - 2576896 \) Copy content Toggle raw display
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