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Properties

Label 10.16.a.b
Level $10$
Weight $16$
Character orbit 10.a
Self dual yes
Analytic conductor $14.269$
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] = [10,16,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, 16, 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, 16); N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-128,-918] 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: \(14.2693505100\)
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)
 
\(f(q)\) \(=\) \( q - 128 q^{2} - 918 q^{3} + 16384 q^{4} - 78125 q^{5} + 117504 q^{6} - 953554 q^{7} - 2097152 q^{8} - 13506183 q^{9} + 10000000 q^{10} + 17783232 q^{11} - 15040512 q^{12} + 140533322 q^{13} + 122054912 q^{14}+ \cdots - 240183585723456 q^{99}+O(q^{100}) \) 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 −918.000 16384.0 −78125.0 117504. −953554. −2.09715e6 −1.35062e7 1.00000e7
\(n\): 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 10.16.a.b ✓ 1
3.b odd 2 1 90.16.a.h 1
4.b odd 2 1 80.16.a.a 1
5.b even 2 1 50.16.a.c 1
5.c odd 4 2 50.16.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.a.b ✓ 1 1.a even 1 1 trivial
50.16.a.c 1 5.b even 2 1
50.16.b.c 2 5.c odd 4 2
80.16.a.a 1 4.b odd 2 1
90.16.a.h 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 918 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 128 \) Copy content Toggle raw display
$3$ \( T + 918 \) Copy content Toggle raw display
$5$ \( T + 78125 \) Copy content Toggle raw display
$7$ \( T + 953554 \) Copy content Toggle raw display
$11$ \( T - 17783232 \) Copy content Toggle raw display
$13$ \( T - 140533322 \) Copy content Toggle raw display
$17$ \( T - 2998870746 \) Copy content Toggle raw display
$19$ \( T - 3255852500 \) Copy content Toggle raw display
$23$ \( T - 6774812202 \) Copy content Toggle raw display
$29$ \( T + 7340322690 \) Copy content Toggle raw display
$31$ \( T + 115428411388 \) Copy content Toggle raw display
$37$ \( T - 150300986906 \) Copy content Toggle raw display
$41$ \( T - 1841603525142 \) Copy content Toggle raw display
$43$ \( T - 1510018315682 \) Copy content Toggle raw display
$47$ \( T - 6093750843366 \) Copy content Toggle raw display
$53$ \( T + 8267412829038 \) Copy content Toggle raw display
$59$ \( T + 23516883061980 \) Copy content Toggle raw display
$61$ \( T + 3135369104278 \) Copy content Toggle raw display
$67$ \( T + 36030983954794 \) Copy content Toggle raw display
$71$ \( T - 52169735384172 \) Copy content Toggle raw display
$73$ \( T - 69977143684082 \) Copy content Toggle raw display
$79$ \( T + 135317670906760 \) Copy content Toggle raw display
$83$ \( T - 427456158822882 \) Copy content Toggle raw display
$89$ \( T + 446581617299190 \) Copy content Toggle raw display
$97$ \( T - 181247411845826 \) Copy content Toggle raw display
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