Properties

Label 3584.2.a.d
Level $3584$
Weight $2$
Character orbit 3584.a
Self dual yes
Analytic conductor $28.618$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + q^{7} + (\beta_{2} - 1) q^{9} - 2 \beta_{3} q^{11} - 3 \beta_1 q^{13} + \beta_{2} q^{15} + ( - 2 \beta_{2} - 2) q^{17} + ( - \beta_{3} + 2 \beta_1) q^{19} + \beta_1 q^{21} + ( - \beta_{2} - 2) q^{23} + ( - \beta_{2} - 3) q^{25} + (\beta_{3} - 3 \beta_1) q^{27} + (4 \beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{2} - 4) q^{31} - 2 \beta_{2} q^{33} + \beta_{3} q^{35} + ( - 2 \beta_{3} - 4 \beta_1) q^{37} + ( - 3 \beta_{2} - 6) q^{39} + (6 \beta_{2} + 2) q^{41} + 2 \beta_1 q^{43} + ( - 2 \beta_{3} + \beta_1) q^{45} - 4 q^{47} + q^{49} + ( - 2 \beta_{3} - 4 \beta_1) q^{51} - 4 \beta_{3} q^{53} + (2 \beta_{2} - 4) q^{55} + (\beta_{2} + 4) q^{57} + ( - 9 \beta_{3} - 2 \beta_1) q^{59} - \beta_{3} q^{61} + (\beta_{2} - 1) q^{63} - 3 \beta_{2} q^{65} + (6 \beta_{3} + 6 \beta_1) q^{67} + ( - \beta_{3} - 3 \beta_1) q^{69} + (2 \beta_{2} - 2) q^{71} + (6 \beta_{2} - 2) q^{73} + ( - \beta_{3} - 4 \beta_1) q^{75} - 2 \beta_{3} q^{77} + (2 \beta_{2} - 6) q^{79} + ( - 5 \beta_{2} - 3) q^{81} - 3 \beta_1 q^{83} - 2 \beta_1 q^{85} + (2 \beta_{2} - 4) q^{87} + 6 q^{89} - 3 \beta_1 q^{91} + ( - 2 \beta_{3} - 6 \beta_1) q^{93} + (3 \beta_{2} - 2) q^{95} - 2 q^{97} + (4 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} - 4 q^{9} - 8 q^{17} - 8 q^{23} - 12 q^{25} - 16 q^{31} - 24 q^{39} + 8 q^{41} - 16 q^{47} + 4 q^{49} - 16 q^{55} + 16 q^{57} - 4 q^{63} - 8 q^{71} - 8 q^{73} - 24 q^{79} - 12 q^{81} - 16 q^{87} + 24 q^{89} - 8 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display

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
−1.84776
−0.765367
0.765367
1.84776
0 −1.84776 0 −0.765367 0 1.00000 0 0.414214 0
1.2 0 −0.765367 0 1.84776 0 1.00000 0 −2.41421 0
1.3 0 0.765367 0 −1.84776 0 1.00000 0 −2.41421 0
1.4 0 1.84776 0 0.765367 0 1.00000 0 0.414214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 3584.2.a.d yes 4
4.b odd 2 1 3584.2.a.c 4
8.b even 2 1 inner 3584.2.a.d yes 4
8.d odd 2 1 3584.2.a.c 4
16.e even 4 2 3584.2.b.c 4
16.f odd 4 2 3584.2.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.c 4 4.b odd 2 1
3584.2.a.c 4 8.d odd 2 1
3584.2.a.d yes 4 1.a even 1 1 trivial
3584.2.a.d yes 4 8.b even 2 1 inner
3584.2.b.c 4 16.e even 4 2
3584.2.b.d 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3584))\):

\( T_{3}^{4} - 4T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{5}^{4} - 4T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 4T_{23} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$5$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 16T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{4} - 36T^{2} + 162 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 20T^{2} + 98 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 80T^{2} + 32 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 80T^{2} + 32 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 16T^{2} + 32 \) Copy content Toggle raw display
$47$ \( (T + 4)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 64T^{2} + 512 \) Copy content Toggle raw display
$59$ \( T^{4} - 340 T^{2} + 25538 \) Copy content Toggle raw display
$61$ \( T^{4} - 4T^{2} + 2 \) Copy content Toggle raw display
$67$ \( T^{4} - 288 T^{2} + 10368 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4 T - 68)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 36T^{2} + 162 \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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