↑

Properties

Label 20.37.d.b
Level $20$
Weight $37$
Character orbit 20.d
Self dual yes
Analytic conductor $164.183$
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] = [20,37,Mod(19,20)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
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, 37, names="a")
 
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, 37); N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 37 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,262144] 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: \(164.182517319\)
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)
 
\(f(q)\) \(=\) \( q + 262144 q^{2} - 652871042 q^{3} + 68719476736 q^{4} + 3814697265625 q^{5} - 171146226434048 q^{6} - 15\!\cdots\!82 q^{7} + 18\!\cdots\!84 q^{8} + 27\!\cdots\!43 q^{9} + 10\!\cdots\!00 q^{10} - 44\!\cdots\!12 q^{12}+ \cdots - 48\!\cdots\!88 q^{98}+O(q^{100}) \) 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
262144. −6.52871e8 6.87195e10 3.81470e12 −1.71146e14 −1.57038e15 1.80144e16 2.76146e17 1.00000e18
\(n\): e.g. 2-40 or 990-1000
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 20.37.d.b yes 1
4.b odd 2 1 20.37.d.a ✓ 1
5.b even 2 1 20.37.d.a ✓ 1
20.d odd 2 1 CM 20.37.d.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.37.d.a ✓ 1 4.b odd 2 1
20.37.d.a ✓ 1 5.b even 2 1
20.37.d.b yes 1 1.a even 1 1 trivial
20.37.d.b yes 1 20.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 652871042 \) acting on \(S_{37}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 262144 \) Copy content Toggle raw display
$3$ \( T + 652871042 \) Copy content Toggle raw display
$5$ \( T - 3814697265625 \) Copy content Toggle raw display
$7$ \( T + 1570377658329682 \) 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
$23$ \( T - 37\!\cdots\!58 \) Copy content Toggle raw display
$29$ \( T + 31\!\cdots\!58 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 20\!\cdots\!62 \) Copy content Toggle raw display
$43$ \( T + 39\!\cdots\!22 \) Copy content Toggle raw display
$47$ \( T + 18\!\cdots\!42 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 23\!\cdots\!42 \) Copy content Toggle raw display
$67$ \( T + 14\!\cdots\!82 \) 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
$83$ \( T - 45\!\cdots\!38 \) Copy content Toggle raw display
$89$ \( T - 14\!\cdots\!42 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
show more
show less