Properties

Label 14.10.a.b
Level $14$
Weight $10$
Character orbit 14.a
Self dual yes
Analytic conductor $7.211$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [14,10,Mod(1,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.21050170629\)
Analytic rank: \(0\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 16 q^{2} + 170 q^{3} + 256 q^{4} + 544 q^{5} + 2720 q^{6} - 2401 q^{7} + 4096 q^{8} + 9217 q^{9} + 8704 q^{10} + 48824 q^{11} + 43520 q^{12} - 15876 q^{13} - 38416 q^{14} + 92480 q^{15} + 65536 q^{16}+ \cdots + 450010808 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
16.0000 170.000 256.000 544.000 2720.00 −2401.00 4096.00 9217.00 8704.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)

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 14.10.a.b 1
3.b odd 2 1 126.10.a.a 1
4.b odd 2 1 112.10.a.a 1
5.b even 2 1 350.10.a.a 1
5.c odd 4 2 350.10.c.d 2
7.b odd 2 1 98.10.a.b 1
7.c even 3 2 98.10.c.a 2
7.d odd 6 2 98.10.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.b 1 1.a even 1 1 trivial
98.10.a.b 1 7.b odd 2 1
98.10.c.a 2 7.c even 3 2
98.10.c.d 2 7.d odd 6 2
112.10.a.a 1 4.b odd 2 1
126.10.a.a 1 3.b odd 2 1
350.10.a.a 1 5.b even 2 1
350.10.c.d 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 170 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(14))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T - 170 \) Copy content Toggle raw display
$5$ \( T - 544 \) Copy content Toggle raw display
$7$ \( T + 2401 \) Copy content Toggle raw display
$11$ \( T - 48824 \) Copy content Toggle raw display
$13$ \( T + 15876 \) Copy content Toggle raw display
$17$ \( T + 21418 \) Copy content Toggle raw display
$19$ \( T + 716410 \) Copy content Toggle raw display
$23$ \( T + 2470000 \) Copy content Toggle raw display
$29$ \( T - 5556826 \) Copy content Toggle raw display
$31$ \( T - 5799348 \) Copy content Toggle raw display
$37$ \( T + 3894430 \) Copy content Toggle raw display
$41$ \( T + 6360858 \) Copy content Toggle raw display
$43$ \( T + 18701296 \) Copy content Toggle raw display
$47$ \( T - 56539068 \) Copy content Toggle raw display
$53$ \( T + 59894682 \) Copy content Toggle raw display
$59$ \( T - 165629662 \) Copy content Toggle raw display
$61$ \( T - 51419016 \) Copy content Toggle raw display
$67$ \( T - 93546508 \) Copy content Toggle raw display
$71$ \( T + 95633536 \) Copy content Toggle raw display
$73$ \( T - 306496402 \) Copy content Toggle raw display
$79$ \( T - 496474152 \) Copy content Toggle raw display
$83$ \( T + 371486962 \) Copy content Toggle raw display
$89$ \( T + 165482550 \) Copy content Toggle raw display
$97$ \( T - 758016742 \) Copy content Toggle raw display
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