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Properties

Label 4160.2.a.bb
Level $4160$
Weight $2$
Character orbit 4160.a
Self dual yes
Analytic conductor $33.218$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4160,2,Mod(1,4160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4160, 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("4160.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4160.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0,0,-6,0,0,0,-2,0,0,0,-12,0,0,0,0,0,0,0,2,0,0,0, 4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(33.2177672409\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2080)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \beta q^{7} - 3 q^{9} + \beta q^{11} - q^{13} - 6 q^{17} + 3 \beta q^{19} + 2 \beta q^{23} + q^{25} + 2 q^{29} - 3 \beta q^{31} - \beta q^{35} + 10 q^{37} - 6 q^{41} - 2 \beta q^{43} + 3 q^{45} + \cdots - 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 6 q^{9} - 2 q^{13} - 12 q^{17} + 2 q^{25} + 4 q^{29} + 20 q^{37} - 12 q^{41} + 6 q^{45} + 2 q^{49} + 4 q^{53} + 20 q^{61} + 2 q^{65} + 20 q^{73} + 16 q^{77} + 18 q^{81} + 12 q^{85} - 12 q^{89}+ \cdots + 4 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 0 0 −1.00000 0 −2.82843 0 −3.00000 0
1.2 0 0 0 −1.00000 0 2.82843 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(13\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4160.2.a.bb 2
4.b odd 2 1 inner 4160.2.a.bb 2
8.b even 2 1 2080.2.a.i ✓ 2
8.d odd 2 1 2080.2.a.i ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2080.2.a.i ✓ 2 8.b even 2 1
2080.2.a.i ✓ 2 8.d odd 2 1
4160.2.a.bb 2 1.a even 1 1 trivial
4160.2.a.bb 2 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4160))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{19}^{2} - 72 \) 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
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 72 \) Copy content Toggle raw display
$23$ \( T^{2} - 32 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 72 \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 32 \) Copy content Toggle raw display
$47$ \( T^{2} - 8 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 72 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 72 \) Copy content Toggle raw display
$71$ \( T^{2} - 8 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 288 \) Copy content Toggle raw display
$83$ \( T^{2} - 200 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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