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Properties

Label 14.14.a.a
Level $14$
Weight $14$
Character orbit 14.a
Self dual yes
Analytic conductor $15.012$
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] = [14,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-64] 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: \(15.0123300533\)
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)
 
\(f(q)\) \(=\) \( q - 64 q^{2} + 1626 q^{3} + 4096 q^{4} - 36400 q^{5} - 104064 q^{6} - 117649 q^{7} - 262144 q^{8} + 1049553 q^{9} + 2329600 q^{10} + 2605288 q^{11} + 6660096 q^{12} - 12624468 q^{13} + 7529536 q^{14} - 59186400 q^{15}+ \cdots + 2734387836264 q^{99}+O(q^{100}) \) 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
−64.0000 1626.00 4096.00 −36400.0 −104064. −117649. −262144. 1.04955e6 2.32960e6
\(n\): 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 14.14.a.a ✓ 1
3.b odd 2 1 126.14.a.e 1
4.b odd 2 1 112.14.a.a 1
7.b odd 2 1 98.14.a.a 1
7.c even 3 2 98.14.c.f 2
7.d odd 6 2 98.14.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.14.a.a ✓ 1 1.a even 1 1 trivial
98.14.a.a 1 7.b odd 2 1
98.14.c.f 2 7.c even 3 2
98.14.c.g 2 7.d odd 6 2
112.14.a.a 1 4.b odd 2 1
126.14.a.e 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1626 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(14))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 64 \) Copy content Toggle raw display
$3$ \( T - 1626 \) Copy content Toggle raw display
$5$ \( T + 36400 \) Copy content Toggle raw display
$7$ \( T + 117649 \) Copy content Toggle raw display
$11$ \( T - 2605288 \) Copy content Toggle raw display
$13$ \( T + 12624468 \) Copy content Toggle raw display
$17$ \( T + 130752362 \) Copy content Toggle raw display
$19$ \( T + 249436042 \) Copy content Toggle raw display
$23$ \( T - 489054160 \) Copy content Toggle raw display
$29$ \( T + 112115926 \) Copy content Toggle raw display
$31$ \( T + 9103068684 \) Copy content Toggle raw display
$37$ \( T - 18308169938 \) Copy content Toggle raw display
$41$ \( T - 13082373606 \) Copy content Toggle raw display
$43$ \( T + 67123460032 \) Copy content Toggle raw display
$47$ \( T - 105239980284 \) Copy content Toggle raw display
$53$ \( T + 25221720042 \) Copy content Toggle raw display
$59$ \( T + 276774602098 \) Copy content Toggle raw display
$61$ \( T - 759388645560 \) Copy content Toggle raw display
$67$ \( T - 1039664575708 \) Copy content Toggle raw display
$71$ \( T - 1817086195456 \) Copy content Toggle raw display
$73$ \( T - 400342248850 \) Copy content Toggle raw display
$79$ \( T + 3597798513336 \) Copy content Toggle raw display
$83$ \( T + 1309030493954 \) Copy content Toggle raw display
$89$ \( T - 1653288354570 \) Copy content Toggle raw display
$97$ \( T + 12736909073690 \) Copy content Toggle raw display
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