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Properties

Label 1584.4.a.bf
Level $1584$
Weight $4$
Character orbit 1584.a
Self dual yes
Analytic conductor $93.459$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1584,4,Mod(1,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1584.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,6,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(93.4590254491\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 264)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 3) q^{5} + ( - \beta - 5) q^{7} + 11 q^{11} + (17 \beta - 1) q^{13} + (18 \beta + 52) q^{17} + (30 \beta - 2) q^{19} + (13 \beta - 51) q^{23} + (6 \beta - 99) q^{25} + ( - 26 \beta + 196) q^{29}+ \cdots + (38 \beta + 984) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} - 10 q^{7} + 22 q^{11} - 2 q^{13} + 104 q^{17} - 4 q^{19} - 102 q^{23} - 198 q^{25} + 392 q^{29} + 64 q^{31} - 64 q^{35} - 164 q^{37} + 732 q^{41} - 168 q^{43} - 314 q^{47} - 602 q^{49} + 382 q^{53}+ \cdots + 1968 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.56155
2.56155
0 0 0 −1.12311 0 −0.876894 0 0 0
1.2 0 0 0 7.12311 0 −9.12311 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(11\) \( -1 \)

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 1584.4.a.bf 2
3.b odd 2 1 528.4.a.r 2
4.b odd 2 1 792.4.a.h 2
12.b even 2 1 264.4.a.e ✓ 2
24.f even 2 1 2112.4.a.bl 2
24.h odd 2 1 2112.4.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.e ✓ 2 12.b even 2 1
528.4.a.r 2 3.b odd 2 1
792.4.a.h 2 4.b odd 2 1
1584.4.a.bf 2 1.a even 1 1 trivial
2112.4.a.be 2 24.h odd 2 1
2112.4.a.bl 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1584))\):

\( T_{5}^{2} - 6T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 10T_{7} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 4912 \) Copy content Toggle raw display
$17$ \( T^{2} - 104T - 2804 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 15296 \) Copy content Toggle raw display
$23$ \( T^{2} + 102T - 272 \) Copy content Toggle raw display
$29$ \( T^{2} - 392T + 26924 \) Copy content Toggle raw display
$31$ \( T^{2} - 64T - 26176 \) Copy content Toggle raw display
$37$ \( T^{2} + 164T - 102076 \) Copy content Toggle raw display
$41$ \( T^{2} - 732T + 127156 \) Copy content Toggle raw display
$43$ \( T^{2} + 168T - 71552 \) Copy content Toggle raw display
$47$ \( T^{2} + 314T - 9776 \) Copy content Toggle raw display
$53$ \( T^{2} - 382T + 13208 \) Copy content Toggle raw display
$59$ \( T^{2} - 508T + 7328 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 387608 \) Copy content Toggle raw display
$67$ \( T^{2} + 216T - 302768 \) Copy content Toggle raw display
$71$ \( T^{2} + 878T + 125248 \) Copy content Toggle raw display
$73$ \( T^{2} - 260T - 836092 \) Copy content Toggle raw display
$79$ \( T^{2} + 118T - 156472 \) Copy content Toggle raw display
$83$ \( T^{2} - 496T - 702544 \) Copy content Toggle raw display
$89$ \( T^{2} - 1756 T + 291076 \) Copy content Toggle raw display
$97$ \( T^{2} - 1968 T + 943708 \) Copy content Toggle raw display
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