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Properties

Label 1296.4.a.p
Level $1296$
Weight $4$
Character orbit 1296.a
Self dual yes
Analytic conductor $76.466$
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] = [1296,4,Mod(1,1296)] 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("1296.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,4,0,6] 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: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 648)
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 \(\beta = \sqrt{129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 2) q^{5} + (\beta + 3) q^{7} + ( - 3 \beta - 5) q^{11} + ( - 5 \beta - 20) q^{13} + ( - 4 \beta - 17) q^{17} + (5 \beta - 17) q^{19} + (7 \beta + 49) q^{23} + (4 \beta + 8) q^{25} + (11 \beta - 60) q^{29}+ \cdots + ( - 122 \beta - 100) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 6 q^{7} - 10 q^{11} - 40 q^{13} - 34 q^{17} - 34 q^{19} + 98 q^{23} + 16 q^{25} - 120 q^{29} + 200 q^{31} + 270 q^{35} + 480 q^{37} - 192 q^{41} + 334 q^{43} + 300 q^{47} - 410 q^{49} - 68 q^{53}+ \cdots - 200 q^{97}+O(q^{100}) \) 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
−5.17891
6.17891
0 0 0 −9.35782 0 −8.35782 0 0 0
1.2 0 0 0 13.3578 0 14.3578 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1296.4.a.p 2
3.b odd 2 1 1296.4.a.n 2
4.b odd 2 1 648.4.a.e yes 2
12.b even 2 1 648.4.a.d ✓ 2
36.f odd 6 2 648.4.i.p 4
36.h even 6 2 648.4.i.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.d ✓ 2 12.b even 2 1
648.4.a.e yes 2 4.b odd 2 1
648.4.i.p 4 36.f odd 6 2
648.4.i.q 4 36.h even 6 2
1296.4.a.n 2 3.b odd 2 1
1296.4.a.p 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 4T_{5} - 125 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\). 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
$5$ \( T^{2} - 4T - 125 \) Copy content Toggle raw display
$7$ \( T^{2} - 6T - 120 \) Copy content Toggle raw display
$11$ \( T^{2} + 10T - 1136 \) Copy content Toggle raw display
$13$ \( T^{2} + 40T - 2825 \) Copy content Toggle raw display
$17$ \( T^{2} + 34T - 1775 \) Copy content Toggle raw display
$19$ \( T^{2} + 34T - 2936 \) Copy content Toggle raw display
$23$ \( T^{2} - 98T - 3920 \) Copy content Toggle raw display
$29$ \( T^{2} + 120T - 12009 \) Copy content Toggle raw display
$31$ \( T^{2} - 200T + 1744 \) Copy content Toggle raw display
$37$ \( T^{2} - 480T + 51279 \) Copy content Toggle raw display
$41$ \( T^{2} + 192T - 3684 \) Copy content Toggle raw display
$43$ \( T^{2} - 334T - 52736 \) Copy content Toggle raw display
$47$ \( T^{2} - 300T - 163776 \) Copy content Toggle raw display
$53$ \( T^{2} + 68T - 130940 \) Copy content Toggle raw display
$59$ \( T^{2} - 620T + 91456 \) Copy content Toggle raw display
$61$ \( T^{2} - 200T - 148025 \) Copy content Toggle raw display
$67$ \( T^{2} - 1406 T + 478600 \) Copy content Toggle raw display
$71$ \( T^{2} - 1390 T + 481864 \) Copy content Toggle raw display
$73$ \( T^{2} - 1802 T + 770005 \) Copy content Toggle raw display
$79$ \( T^{2} - 334T - 777200 \) Copy content Toggle raw display
$83$ \( T^{2} + 500T - 643904 \) Copy content Toggle raw display
$89$ \( T^{2} + 90T - 72279 \) Copy content Toggle raw display
$97$ \( T^{2} + 200 T - 1910036 \) Copy content Toggle raw display
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