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Properties

Label 240.6.a.n
Level $240$
Weight $6$
Character orbit 240.a
Self dual yes
Analytic conductor $38.492$
Analytic rank $1$
Dimension $1$
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] = [240,6,Mod(1,240)] 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("240.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,9,0,25,0,80] 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: \(38.4921167551\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} + 25 q^{5} + 80 q^{7} + 81 q^{9} - 684 q^{11} - 978 q^{13} + 225 q^{15} - 862 q^{17} - 916 q^{19} + 720 q^{21} + 1552 q^{23} + 625 q^{25} + 729 q^{27} - 7314 q^{29} + 9312 q^{31} - 6156 q^{33}+ \cdots - 55404 q^{99}+O(q^{100}) \) 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
0
0 9.00000 0 25.0000 0 80.0000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)

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 240.6.a.n 1
3.b odd 2 1 720.6.a.g 1
4.b odd 2 1 120.6.a.c ✓ 1
8.b even 2 1 960.6.a.f 1
8.d odd 2 1 960.6.a.o 1
12.b even 2 1 360.6.a.c 1
20.d odd 2 1 600.6.a.g 1
20.e even 4 2 600.6.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.a.c ✓ 1 4.b odd 2 1
240.6.a.n 1 1.a even 1 1 trivial
360.6.a.c 1 12.b even 2 1
600.6.a.g 1 20.d odd 2 1
600.6.f.i 2 20.e even 4 2
720.6.a.g 1 3.b odd 2 1
960.6.a.f 1 8.b even 2 1
960.6.a.o 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 80 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(240))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T - 80 \) Copy content Toggle raw display
$11$ \( T + 684 \) Copy content Toggle raw display
$13$ \( T + 978 \) Copy content Toggle raw display
$17$ \( T + 862 \) Copy content Toggle raw display
$19$ \( T + 916 \) Copy content Toggle raw display
$23$ \( T - 1552 \) Copy content Toggle raw display
$29$ \( T + 7314 \) Copy content Toggle raw display
$31$ \( T - 9312 \) Copy content Toggle raw display
$37$ \( T + 8826 \) Copy content Toggle raw display
$41$ \( T + 3286 \) Copy content Toggle raw display
$43$ \( T + 7556 \) Copy content Toggle raw display
$47$ \( T - 5960 \) Copy content Toggle raw display
$53$ \( T + 8698 \) Copy content Toggle raw display
$59$ \( T - 42036 \) Copy content Toggle raw display
$61$ \( T - 37518 \) Copy content Toggle raw display
$67$ \( T + 29324 \) Copy content Toggle raw display
$71$ \( T + 84408 \) Copy content Toggle raw display
$73$ \( T + 46550 \) Copy content Toggle raw display
$79$ \( T + 26752 \) Copy content Toggle raw display
$83$ \( T - 7956 \) Copy content Toggle raw display
$89$ \( T - 59674 \) Copy content Toggle raw display
$97$ \( T - 136898 \) Copy content Toggle raw display
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