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Properties

Label 14.16.a.a
Level $14$
Weight $16$
Character orbit 14.a
Self dual yes
Analytic conductor $19.977$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [14,16,Mod(1,14)] 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("14.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-128] 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: \(19.9770907140\)
Analytic rank: \(1\)
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

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)
 
\(f(q)\) \(=\) \( q - 128 q^{2} + 1350 q^{3} + 16384 q^{4} - 81060 q^{5} - 172800 q^{6} + 823543 q^{7} - 2097152 q^{8} - 12526407 q^{9} + 10375680 q^{10} + 70121184 q^{11} + 22118400 q^{12} + 151469552 q^{13} - 105413504 q^{14}+ \cdots - 878366490105888 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−128.000 1350.00 16384.0 −81060.0 −172800. 823543. −2.09715e6 −1.25264e7 1.03757e7
\(n\): 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 14.16.a.a ✓ 1
3.b odd 2 1 126.16.a.e 1
4.b odd 2 1 112.16.a.a 1
7.b odd 2 1 98.16.a.b 1
7.c even 3 2 98.16.c.b 2
7.d odd 6 2 98.16.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.16.a.a ✓ 1 1.a even 1 1 trivial
98.16.a.b 1 7.b odd 2 1
98.16.c.b 2 7.c even 3 2
98.16.c.c 2 7.d odd 6 2
112.16.a.a 1 4.b odd 2 1
126.16.a.e 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1350 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(14))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 128 \) Copy content Toggle raw display
$3$ \( T - 1350 \) Copy content Toggle raw display
$5$ \( T + 81060 \) Copy content Toggle raw display
$7$ \( T - 823543 \) Copy content Toggle raw display
$11$ \( T - 70121184 \) Copy content Toggle raw display
$13$ \( T - 151469552 \) Copy content Toggle raw display
$17$ \( T + 249756546 \) Copy content Toggle raw display
$19$ \( T + 6476856550 \) Copy content Toggle raw display
$23$ \( T + 21129196200 \) Copy content Toggle raw display
$29$ \( T - 7794825354 \) Copy content Toggle raw display
$31$ \( T + 95032053412 \) Copy content Toggle raw display
$37$ \( T + 870082295470 \) Copy content Toggle raw display
$41$ \( T - 1007666657262 \) Copy content Toggle raw display
$43$ \( T - 155007585272 \) Copy content Toggle raw display
$47$ \( T + 2551970135004 \) Copy content Toggle raw display
$53$ \( T - 4047645687774 \) Copy content Toggle raw display
$59$ \( T + 12599248786302 \) Copy content Toggle raw display
$61$ \( T + 39925031318044 \) Copy content Toggle raw display
$67$ \( T + 48423780261124 \) Copy content Toggle raw display
$71$ \( T - 37693101366144 \) Copy content Toggle raw display
$73$ \( T - 141416194574306 \) Copy content Toggle raw display
$79$ \( T - 247020521013128 \) Copy content Toggle raw display
$83$ \( T - 2788789610034 \) Copy content Toggle raw display
$89$ \( T + 5839634731110 \) Copy content Toggle raw display
$97$ \( T - 278027158065374 \) Copy content Toggle raw display
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