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Properties

Label 30.16.a.b
Level $30$
Weight $16$
Character orbit 30.a
Self dual yes
Analytic conductor $42.808$
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] = [30,16,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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-128,-2187,16384,78125] 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: \(42.8080515300\)
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)
 
\(f(q)\) \(=\) \( q - 128 q^{2} - 2187 q^{3} + 16384 q^{4} + 78125 q^{5} + 279936 q^{6} - 1107904 q^{7} - 2097152 q^{8} + 4782969 q^{9} - 10000000 q^{10} - 4881228 q^{11} - 35831808 q^{12} - 188323018 q^{13} + 141811712 q^{14}+ \cdots - 23346762205932 q^{99}+O(q^{100}) \) 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 −2187.00 16384.0 78125.0 279936. −1.10790e6 −2.09715e6 4.78297e6 −1.00000e7
\(n\): 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 30.16.a.b ✓ 1
3.b odd 2 1 90.16.a.e 1
5.b even 2 1 150.16.a.j 1
5.c odd 4 2 150.16.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.16.a.b ✓ 1 1.a even 1 1 trivial
90.16.a.e 1 3.b odd 2 1
150.16.a.j 1 5.b even 2 1
150.16.c.g 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1107904 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(30))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 128 \) Copy content Toggle raw display
$3$ \( T + 2187 \) Copy content Toggle raw display
$5$ \( T - 78125 \) Copy content Toggle raw display
$7$ \( T + 1107904 \) Copy content Toggle raw display
$11$ \( T + 4881228 \) Copy content Toggle raw display
$13$ \( T + 188323018 \) Copy content Toggle raw display
$17$ \( T - 1447226466 \) Copy content Toggle raw display
$19$ \( T - 4465130540 \) Copy content Toggle raw display
$23$ \( T - 1547548992 \) Copy content Toggle raw display
$29$ \( T - 35474823510 \) Copy content Toggle raw display
$31$ \( T - 42074613632 \) Copy content Toggle raw display
$37$ \( T + 160515377794 \) Copy content Toggle raw display
$41$ \( T + 297244163718 \) Copy content Toggle raw display
$43$ \( T - 316105579532 \) Copy content Toggle raw display
$47$ \( T + 4630961371464 \) Copy content Toggle raw display
$53$ \( T + 14006688960738 \) Copy content Toggle raw display
$59$ \( T + 12753794926860 \) Copy content Toggle raw display
$61$ \( T + 17408324211178 \) Copy content Toggle raw display
$67$ \( T - 48459881059076 \) Copy content Toggle raw display
$71$ \( T + 3941529158328 \) Copy content Toggle raw display
$73$ \( T + 97983052629958 \) Copy content Toggle raw display
$79$ \( T - 99129620994560 \) Copy content Toggle raw display
$83$ \( T + 341261170614588 \) Copy content Toggle raw display
$89$ \( T + 5624510252790 \) Copy content Toggle raw display
$97$ \( T + 108753936110014 \) Copy content Toggle raw display
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