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Properties

Label 10.22.a.a
Level $10$
Weight $22$
Character orbit 10.a
Self dual yes
Analytic conductor $27.948$
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] = [10,22,Mod(1,10)] 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("10.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
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, 22, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1024,-21924] 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: \(27.9477344287\)
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)
 
\(f(q)\) \(=\) \( q + 1024 q^{2} - 21924 q^{3} + 1048576 q^{4} + 9765625 q^{5} - 22450176 q^{6} - 722753248 q^{7} + 1073741824 q^{8} - 9979691427 q^{9} + 10000000000 q^{10} + 49976398572 q^{11} - 22988980224 q^{12} - 24351400354 q^{13}+ \cdots - 49\!\cdots\!44 q^{99}+O(q^{100}) \) 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
1024.00 −21924.0 1.04858e6 9.76562e6 −2.24502e7 −7.22753e8 1.07374e9 −9.97969e9 1.00000e10
\(n\): 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 10.22.a.a ✓ 1
4.b odd 2 1 80.22.a.a 1
5.b even 2 1 50.22.a.b 1
5.c odd 4 2 50.22.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.22.a.a ✓ 1 1.a even 1 1 trivial
50.22.a.b 1 5.b even 2 1
50.22.b.b 2 5.c odd 4 2
80.22.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 21924 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1024 \) Copy content Toggle raw display
$3$ \( T + 21924 \) Copy content Toggle raw display
$5$ \( T - 9765625 \) Copy content Toggle raw display
$7$ \( T + 722753248 \) Copy content Toggle raw display
$11$ \( T - 49976398572 \) Copy content Toggle raw display
$13$ \( T + 24351400354 \) Copy content Toggle raw display
$17$ \( T + 5768874283278 \) Copy content Toggle raw display
$19$ \( T + 30250225982620 \) Copy content Toggle raw display
$23$ \( T + 145464074718144 \) Copy content Toggle raw display
$29$ \( T + 1167107530943250 \) Copy content Toggle raw display
$31$ \( T + 7431907384909648 \) Copy content Toggle raw display
$37$ \( T + 54\!\cdots\!98 \) Copy content Toggle raw display
$41$ \( T + 20\!\cdots\!78 \) Copy content Toggle raw display
$43$ \( T - 76\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T - 50\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T - 13\!\cdots\!06 \) Copy content Toggle raw display
$59$ \( T - 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T - 52\!\cdots\!02 \) Copy content Toggle raw display
$67$ \( T + 10\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T + 13\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T + 17\!\cdots\!14 \) Copy content Toggle raw display
$79$ \( T - 11\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T - 13\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T - 34\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T + 37\!\cdots\!98 \) Copy content Toggle raw display
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