↑

Properties

Label 50.8.a.c
Level $50$
Weight $8$
Character orbit 50.a
Self dual yes
Analytic conductor $15.619$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,8,Mod(1,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-8,57] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.6192512742\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} + 57 q^{3} + 64 q^{4} - 456 q^{6} - 1174 q^{7} - 512 q^{8} + 1062 q^{9} - 7563 q^{11} + 3648 q^{12} + 5372 q^{13} + 9392 q^{14} + 4096 q^{16} + 24021 q^{17} - 8496 q^{18} - 51235 q^{19} - 66918 q^{21}+ \cdots - 8031906 q^{99}+O(q^{100}) \) 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
−8.00000 57.0000 64.0000 0 −456.000 −1174.00 −512.000 1062.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 50.8.a.c ✓ 1
3.b odd 2 1 450.8.a.p 1
4.b odd 2 1 400.8.a.d 1
5.b even 2 1 50.8.a.f yes 1
5.c odd 4 2 50.8.b.b 2
15.d odd 2 1 450.8.a.l 1
15.e even 4 2 450.8.c.q 2
20.d odd 2 1 400.8.a.q 1
20.e even 4 2 400.8.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.8.a.c ✓ 1 1.a even 1 1 trivial
50.8.a.f yes 1 5.b even 2 1
50.8.b.b 2 5.c odd 4 2
400.8.a.d 1 4.b odd 2 1
400.8.a.q 1 20.d odd 2 1
400.8.c.d 2 20.e even 4 2
450.8.a.l 1 15.d odd 2 1
450.8.a.p 1 3.b odd 2 1
450.8.c.q 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 57 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(50))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T - 57 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1174 \) Copy content Toggle raw display
$11$ \( T + 7563 \) Copy content Toggle raw display
$13$ \( T - 5372 \) Copy content Toggle raw display
$17$ \( T - 24021 \) Copy content Toggle raw display
$19$ \( T + 51235 \) Copy content Toggle raw display
$23$ \( T + 57618 \) Copy content Toggle raw display
$29$ \( T - 47040 \) Copy content Toggle raw display
$31$ \( T + 192358 \) Copy content Toggle raw display
$37$ \( T - 197066 \) Copy content Toggle raw display
$41$ \( T + 237723 \) Copy content Toggle raw display
$43$ \( T - 653012 \) Copy content Toggle raw display
$47$ \( T + 826884 \) Copy content Toggle raw display
$53$ \( T - 569022 \) Copy content Toggle raw display
$59$ \( T - 1501080 \) Copy content Toggle raw display
$61$ \( T + 2068918 \) Copy content Toggle raw display
$67$ \( T + 3444349 \) Copy content Toggle raw display
$71$ \( T - 4121052 \) Copy content Toggle raw display
$73$ \( T + 83653 \) Copy content Toggle raw display
$79$ \( T - 1454030 \) Copy content Toggle raw display
$83$ \( T - 1626567 \) Copy content Toggle raw display
$89$ \( T - 6004335 \) Copy content Toggle raw display
$97$ \( T - 3411746 \) Copy content Toggle raw display
show more
show less