↑

Properties

Label 20.11.d.a
Level $20$
Weight $11$
Character orbit 20.d
Self dual yes
Analytic conductor $12.707$
Analytic rank $0$
Dimension $1$
CM discriminant -20
Inner twists $2$

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] = [20,11,Mod(19,20)] 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("20.19"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.7071450535\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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)
 
\(f(q)\) \(=\) \( q - 32 q^{2} - 236 q^{3} + 1024 q^{4} - 3125 q^{5} + 7552 q^{6} - 33364 q^{7} - 32768 q^{8} - 3353 q^{9} + 100000 q^{10} - 241664 q^{12} + 1067648 q^{14} + 737500 q^{15} + 1048576 q^{16} + 107296 q^{18}+ \cdots - 26581799904 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−32.0000 −236.000 1024.00 −3125.00 7552.00 −33364.0 −32768.0 −3353.00 100000.
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.11.d.a ✓ 1
4.b odd 2 1 20.11.d.b yes 1
5.b even 2 1 20.11.d.b yes 1
5.c odd 4 2 100.11.b.c 2
20.d odd 2 1 CM 20.11.d.a ✓ 1
20.e even 4 2 100.11.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.d.a ✓ 1 1.a even 1 1 trivial
20.11.d.a ✓ 1 20.d odd 2 1 CM
20.11.d.b yes 1 4.b odd 2 1
20.11.d.b yes 1 5.b even 2 1
100.11.b.c 2 5.c odd 4 2
100.11.b.c 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 236 \) acting on \(S_{11}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 32 \) Copy content Toggle raw display
$3$ \( T + 236 \) Copy content Toggle raw display
$5$ \( T + 3125 \) Copy content Toggle raw display
$7$ \( T + 33364 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 1169564 \) Copy content Toggle raw display
$29$ \( T + 38179702 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 211028098 \) Copy content Toggle raw display
$43$ \( T - 223663364 \) Copy content Toggle raw display
$47$ \( T + 96887764 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 1041591898 \) Copy content Toggle raw display
$67$ \( T + 2343243964 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 5449159036 \) Copy content Toggle raw display
$89$ \( T - 11118190898 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
show more
show less