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Properties

Label 5616.2.a.bz
Level $5616$
Weight $2$
Character orbit 5616.a
Self dual yes
Analytic conductor $44.844$
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] = [5616,2,Mod(1,5616)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5616.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5616 = 2^{4} \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5616.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,5,0,2,0,0,0,-1,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.8439857752\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 351)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{5} + q^{7} - \beta q^{11} + q^{13} + ( - \beta + 3) q^{17} + ( - \beta + 4) q^{19} + 3 \beta q^{23} + ( - 5 \beta + 7) q^{25} + (2 \beta + 3) q^{29} + (4 \beta - 2) q^{31} + ( - \beta + 3) q^{35} + \cdots + (8 \beta - 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{5} + 2 q^{7} - q^{11} + 2 q^{13} + 5 q^{17} + 7 q^{19} + 3 q^{23} + 9 q^{25} + 8 q^{29} + 5 q^{35} - 6 q^{37} + 18 q^{41} - 5 q^{43} - 9 q^{47} - 12 q^{49} - 15 q^{53} + 4 q^{55} - 6 q^{59}+ \cdots + 24 q^{95}+O(q^{100}) \) 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
2.30278
−1.30278
0 0 0 0.697224 0 1.00000 0 0 0
1.2 0 0 0 4.30278 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(13\) \( -1 \)

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 5616.2.a.bz 2
3.b odd 2 1 5616.2.a.bi 2
4.b odd 2 1 351.2.a.d yes 2
12.b even 2 1 351.2.a.b ✓ 2
20.d odd 2 1 8775.2.a.w 2
36.f odd 6 2 1053.2.e.f 4
36.h even 6 2 1053.2.e.j 4
52.b odd 2 1 4563.2.a.i 2
60.h even 2 1 8775.2.a.bd 2
156.h even 2 1 4563.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
351.2.a.b ✓ 2 12.b even 2 1
351.2.a.d yes 2 4.b odd 2 1
1053.2.e.f 4 36.f odd 6 2
1053.2.e.j 4 36.h even 6 2
4563.2.a.i 2 52.b odd 2 1
4563.2.a.q 2 156.h even 2 1
5616.2.a.bi 2 3.b odd 2 1
5616.2.a.bz 2 1.a even 1 1 trivial
8775.2.a.w 2 20.d odd 2 1
8775.2.a.bd 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5616))\):

\( T_{5}^{2} - 5T_{5} + 3 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 5T_{17} + 3 \) 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
$5$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T - 27 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$31$ \( T^{2} - 52 \) Copy content Toggle raw display
$37$ \( T^{2} + 6T - 43 \) Copy content Toggle raw display
$41$ \( (T - 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 3 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 15T + 27 \) Copy content Toggle raw display
$59$ \( (T + 3)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 12T + 23 \) Copy content Toggle raw display
$67$ \( T^{2} - 9T - 61 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T - 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 17T + 43 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T + 3 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$97$ \( T^{2} - 208 \) Copy content Toggle raw display
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