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Properties

Label 30.20.a.b
Level $30$
Weight $20$
Character orbit 30.a
Self dual yes
Analytic conductor $68.645$
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] = [30,20,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, 20); 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, 20, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,512,19683,262144,1953125] 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: \(68.6450089669\)
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)
 
\(f(q)\) \(=\) \( q + 512 q^{2} + 19683 q^{3} + 262144 q^{4} + 1953125 q^{5} + 10077696 q^{6} - 18629464 q^{7} + 134217728 q^{8} + 387420489 q^{9} + 1000000000 q^{10} - 14253161868 q^{11} + 5159780352 q^{12} - 26283085738 q^{13}+ \cdots - 55\!\cdots\!52 q^{99}+O(q^{100}) \) 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
512.000 19683.0 262144. 1.95312e6 1.00777e7 −1.86295e7 1.34218e8 3.87420e8 1.00000e9
\(n\): 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 30.20.a.b ✓ 1
3.b odd 2 1 90.20.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.20.a.b ✓ 1 1.a even 1 1 trivial
90.20.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 18629464 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(30))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 512 \) Copy content Toggle raw display
$3$ \( T - 19683 \) Copy content Toggle raw display
$5$ \( T - 1953125 \) Copy content Toggle raw display
$7$ \( T + 18629464 \) Copy content Toggle raw display
$11$ \( T + 14253161868 \) Copy content Toggle raw display
$13$ \( T + 26283085738 \) Copy content Toggle raw display
$17$ \( T + 261521716494 \) Copy content Toggle raw display
$19$ \( T + 2121940844500 \) Copy content Toggle raw display
$23$ \( T - 1713422915352 \) Copy content Toggle raw display
$29$ \( T + 31196216636250 \) Copy content Toggle raw display
$31$ \( T + 131624595758368 \) Copy content Toggle raw display
$37$ \( T - 1126125043346846 \) Copy content Toggle raw display
$41$ \( T + 785594175939078 \) Copy content Toggle raw display
$43$ \( T + 1233740482269628 \) Copy content Toggle raw display
$47$ \( T - 3272803645139136 \) Copy content Toggle raw display
$53$ \( T - 9350808705822702 \) Copy content Toggle raw display
$59$ \( T - 19\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T - 15\!\cdots\!82 \) Copy content Toggle raw display
$67$ \( T + 25\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T + 15\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T - 36\!\cdots\!02 \) Copy content Toggle raw display
$79$ \( T + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T + 20\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T + 47\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T + 10\!\cdots\!14 \) Copy content Toggle raw display
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