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Properties

Label 30.10.a.f
Level $30$
Weight $10$
Character orbit 30.a
Self dual yes
Analytic conductor $15.451$
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] = [30,10,Mod(1,30)] 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("30.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,16,81,256,625] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.4510750849\)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} + 81 q^{3} + 256 q^{4} + 625 q^{5} + 1296 q^{6} + 2408 q^{7} + 4096 q^{8} + 6561 q^{9} + 10000 q^{10} + 14892 q^{11} + 20736 q^{12} + 30254 q^{13} + 38528 q^{14} + 50625 q^{15} + 65536 q^{16}+ \cdots + 97706412 q^{99}+O(q^{100}) \) 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
16.0000 81.0000 256.000 625.000 1296.00 2408.00 4096.00 6561.00 10000.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)

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 30.10.a.f ✓ 1
3.b odd 2 1 90.10.a.c 1
4.b odd 2 1 240.10.a.b 1
5.b even 2 1 150.10.a.a 1
5.c odd 4 2 150.10.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.10.a.f ✓ 1 1.a even 1 1 trivial
90.10.a.c 1 3.b odd 2 1
150.10.a.a 1 5.b even 2 1
150.10.c.i 2 5.c odd 4 2
240.10.a.b 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 2408 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(30))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T - 625 \) Copy content Toggle raw display
$7$ \( T - 2408 \) Copy content Toggle raw display
$11$ \( T - 14892 \) Copy content Toggle raw display
$13$ \( T - 30254 \) Copy content Toggle raw display
$17$ \( T - 101778 \) Copy content Toggle raw display
$19$ \( T - 160820 \) Copy content Toggle raw display
$23$ \( T - 526584 \) Copy content Toggle raw display
$29$ \( T - 1788030 \) Copy content Toggle raw display
$31$ \( T + 706528 \) Copy content Toggle raw display
$37$ \( T + 8889082 \) Copy content Toggle raw display
$41$ \( T + 10313238 \) Copy content Toggle raw display
$43$ \( T + 27839956 \) Copy content Toggle raw display
$47$ \( T + 54742512 \) Copy content Toggle raw display
$53$ \( T + 101510826 \) Copy content Toggle raw display
$59$ \( T + 118394340 \) Copy content Toggle raw display
$61$ \( T - 178661342 \) Copy content Toggle raw display
$67$ \( T - 244239428 \) Copy content Toggle raw display
$71$ \( T - 81740232 \) Copy content Toggle raw display
$73$ \( T - 277364234 \) Copy content Toggle raw display
$79$ \( T + 140711920 \) Copy content Toggle raw display
$83$ \( T + 422051436 \) Copy content Toggle raw display
$89$ \( T + 753422310 \) Copy content Toggle raw display
$97$ \( T - 1041114338 \) Copy content Toggle raw display
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