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Properties

Label 10.18.a.a
Level $10$
Weight $18$
Character orbit 10.a
Self dual yes
Analytic conductor $18.322$
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] = [10,18,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, 18); 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, 18, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,256,-14976] 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: \(18.3222087345\)
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)
 
\(f(q)\) \(=\) \( q + 256 q^{2} - 14976 q^{3} + 65536 q^{4} + 390625 q^{5} - 3833856 q^{6} + 14808668 q^{7} + 16777216 q^{8} + 95140413 q^{9} + 100000000 q^{10} - 1085034588 q^{11} - 981467136 q^{12} - 4595303746 q^{13}+ \cdots - 10\!\cdots\!44 q^{99}+O(q^{100}) \) 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
256.000 −14976.0 65536.0 390625. −3.83386e6 1.48087e7 1.67772e7 9.51404e7 1.00000e8
\(n\): 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 10.18.a.a ✓ 1
3.b odd 2 1 90.18.a.b 1
4.b odd 2 1 80.18.a.a 1
5.b even 2 1 50.18.a.b 1
5.c odd 4 2 50.18.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.a ✓ 1 1.a even 1 1 trivial
50.18.a.b 1 5.b even 2 1
50.18.b.a 2 5.c odd 4 2
80.18.a.a 1 4.b odd 2 1
90.18.a.b 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 14976 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 256 \) Copy content Toggle raw display
$3$ \( T + 14976 \) Copy content Toggle raw display
$5$ \( T - 390625 \) Copy content Toggle raw display
$7$ \( T - 14808668 \) Copy content Toggle raw display
$11$ \( T + 1085034588 \) Copy content Toggle raw display
$13$ \( T + 4595303746 \) Copy content Toggle raw display
$17$ \( T + 16104698622 \) Copy content Toggle raw display
$19$ \( T - 48093117860 \) Copy content Toggle raw display
$23$ \( T + 571023069276 \) Copy content Toggle raw display
$29$ \( T + 1726424788290 \) Copy content Toggle raw display
$31$ \( T + 5623721940808 \) Copy content Toggle raw display
$37$ \( T + 10013128639162 \) Copy content Toggle raw display
$41$ \( T + 37505113176198 \) Copy content Toggle raw display
$43$ \( T - 136226190448184 \) Copy content Toggle raw display
$47$ \( T + 37681319902812 \) Copy content Toggle raw display
$53$ \( T + 22543738268346 \) Copy content Toggle raw display
$59$ \( T - 221363585667420 \) Copy content Toggle raw display
$61$ \( T + 276238009706818 \) Copy content Toggle raw display
$67$ \( T - 6165400365120968 \) Copy content Toggle raw display
$71$ \( T + 129445634389248 \) Copy content Toggle raw display
$73$ \( T + 9751215737952646 \) Copy content Toggle raw display
$79$ \( T + 25\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T - 29\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T + 24\!\cdots\!10 \) Copy content Toggle raw display
$97$ \( T - 12\!\cdots\!78 \) Copy content Toggle raw display
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