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Properties

Label 20.29.d.b
Level $20$
Weight $29$
Character orbit 20.d
Self dual yes
Analytic conductor $99.337$
Analytic rank $0$
Dimension $1$
CM discriminant -20
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [20,29,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, 29, 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, 29); N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 29 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,16384] 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: \(99.3369484963\)
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)
 
\(f(q)\) \(=\) \( q + 16384 q^{2} + 6723358 q^{3} + 268435456 q^{4} + 6103515625 q^{5} + 110155497472 q^{6} + 827222074478 q^{7} + 4398046511104 q^{8} + 22326750341203 q^{9} + 100000000000000 q^{10} + 18\!\cdots\!48 q^{12}+ \cdots + 36\!\cdots\!72 q^{98}+O(q^{100}) \) 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
16384.0 6.72336e6 2.68435e8 6.10352e9 1.10155e11 8.27222e11 4.39805e12 2.23268e13 1.00000e14
\(n\): 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 20.29.d.b yes 1
4.b odd 2 1 20.29.d.a ✓ 1
5.b even 2 1 20.29.d.a ✓ 1
20.d odd 2 1 CM 20.29.d.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.29.d.a ✓ 1 4.b odd 2 1
20.29.d.a ✓ 1 5.b even 2 1
20.29.d.b yes 1 1.a even 1 1 trivial
20.29.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 \( T_{3} - 6723358 \) acting on \(S_{29}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16384 \) Copy content Toggle raw display
$3$ \( T - 6723358 \) Copy content Toggle raw display
$5$ \( T - 6103515625 \) Copy content Toggle raw display
$7$ \( T - 827222074478 \) 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
$23$ \( T + 12\!\cdots\!02 \) Copy content Toggle raw display
$29$ \( T + 39\!\cdots\!58 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 64\!\cdots\!98 \) Copy content Toggle raw display
$43$ \( T - 12\!\cdots\!18 \) Copy content Toggle raw display
$47$ \( T + 42\!\cdots\!42 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 15\!\cdots\!98 \) Copy content Toggle raw display
$67$ \( T - 36\!\cdots\!58 \) 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
$83$ \( T + 13\!\cdots\!62 \) Copy content Toggle raw display
$89$ \( T + 36\!\cdots\!98 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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