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Properties

Label 15.13.d.b
Level $15$
Weight $13$
Character orbit 15.d
Self dual yes
Analytic conductor $13.710$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

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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] = [15,13,Mod(14,15)] 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("15.14"); S:= CuspForms(chi, 13); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 13, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 15.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,7] 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: \(13.7099072591\)
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)
 
\(f(q)\) \(=\) \( q + 7 q^{2} - 729 q^{3} - 4047 q^{4} - 15625 q^{5} - 5103 q^{6} - 57001 q^{8} + 531441 q^{9} - 109375 q^{10} + 2950263 q^{12} + 11390625 q^{15} + 16177505 q^{16} - 39972098 q^{17} + 3720087 q^{18} + 79567922 q^{19}+ \cdots + 96889010407 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
0
7.00000 −729.000 −4047.00 −15625.0 −5103.00 0 −57001.0 531441. −109375.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.13.d.b yes 1
3.b odd 2 1 15.13.d.a ✓ 1
5.b even 2 1 15.13.d.a ✓ 1
5.c odd 4 2 75.13.c.b 2
15.d odd 2 1 CM 15.13.d.b yes 1
15.e even 4 2 75.13.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.13.d.a ✓ 1 3.b odd 2 1
15.13.d.a ✓ 1 5.b even 2 1
15.13.d.b yes 1 1.a even 1 1 trivial
15.13.d.b yes 1 15.d odd 2 1 CM
75.13.c.b 2 5.c odd 4 2
75.13.c.b 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 7 \) acting on \(S_{13}^{\mathrm{new}}(15, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 7 \) Copy content Toggle raw display
$3$ \( T + 729 \) Copy content Toggle raw display
$5$ \( T + 15625 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 39972098 \) Copy content Toggle raw display
$19$ \( T - 79567922 \) Copy content Toggle raw display
$23$ \( T - 81332062 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1741852798 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 13452309502 \) Copy content Toggle raw display
$53$ \( T - 36467062702 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 3103588082 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 312456859202 \) Copy content Toggle raw display
$83$ \( T - 433407300622 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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