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Properties

Label 10.14.a.b
Level $10$
Weight $14$
Character orbit 10.a
Self dual yes
Analytic conductor $10.723$
Analytic rank $0$
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] = [10,14,Mod(1,10)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("10.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-64,1224] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.7230928952\)
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

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)
 
\(f(q)\) \(=\) \( q - 64 q^{2} + 1224 q^{3} + 4096 q^{4} + 15625 q^{5} - 78336 q^{6} - 65212 q^{7} - 262144 q^{8} - 96147 q^{9} - 1000000 q^{10} + 7427652 q^{11} + 5013504 q^{12} + 32243054 q^{13} + 4173568 q^{14} + 19125000 q^{15}+ \cdots - 714146456844 q^{99}+O(q^{100}) \) 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 1224.00 4096.00 15625.0 −78336.0 −65212.0 −262144. −96147.0 −1.00000e6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)

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 10.14.a.b ✓ 1
3.b odd 2 1 90.14.a.e 1
4.b odd 2 1 80.14.a.a 1
5.b even 2 1 50.14.a.c 1
5.c odd 4 2 50.14.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.14.a.b ✓ 1 1.a even 1 1 trivial
50.14.a.c 1 5.b even 2 1
50.14.b.b 2 5.c odd 4 2
80.14.a.a 1 4.b odd 2 1
90.14.a.e 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1224 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 64 \) Copy content Toggle raw display
$3$ \( T - 1224 \) Copy content Toggle raw display
$5$ \( T - 15625 \) Copy content Toggle raw display
$7$ \( T + 65212 \) Copy content Toggle raw display
$11$ \( T - 7427652 \) Copy content Toggle raw display
$13$ \( T - 32243054 \) Copy content Toggle raw display
$17$ \( T + 20088222 \) Copy content Toggle raw display
$19$ \( T - 77070740 \) Copy content Toggle raw display
$23$ \( T - 664071804 \) Copy content Toggle raw display
$29$ \( T - 1558250670 \) Copy content Toggle raw display
$31$ \( T + 303290968 \) Copy content Toggle raw display
$37$ \( T + 775029322 \) Copy content Toggle raw display
$41$ \( T - 43696205082 \) Copy content Toggle raw display
$43$ \( T + 68680553536 \) Copy content Toggle raw display
$47$ \( T + 138979393812 \) Copy content Toggle raw display
$53$ \( T + 103656826986 \) Copy content Toggle raw display
$59$ \( T - 394887188940 \) Copy content Toggle raw display
$61$ \( T + 488570895538 \) Copy content Toggle raw display
$67$ \( T - 368381730848 \) Copy content Toggle raw display
$71$ \( T - 325473704592 \) Copy content Toggle raw display
$73$ \( T + 2262556998406 \) Copy content Toggle raw display
$79$ \( T - 2032917332000 \) Copy content Toggle raw display
$83$ \( T + 854518199496 \) Copy content Toggle raw display
$89$ \( T - 8906829484890 \) Copy content Toggle raw display
$97$ \( T + 9873550533742 \) Copy content Toggle raw display
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