↑

Properties

Label 30.14.a.f
Level $30$
Weight $14$
Character orbit 30.a
Self dual yes
Analytic conductor $32.169$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [30,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 30.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,64,729,4096,-15625] 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: \(32.1692786856\)
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)
 
\(f(q)\) \(=\) \( q + 64 q^{2} + 729 q^{3} + 4096 q^{4} - 15625 q^{5} + 46656 q^{6} - 246868 q^{7} + 262144 q^{8} + 531441 q^{9} - 1000000 q^{10} - 5813808 q^{11} + 2985984 q^{12} - 13914094 q^{13} - 15799552 q^{14} - 11390625 q^{15}+ \cdots - 3089695937328 q^{99}+O(q^{100}) \) 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
64.0000 729.000 4096.00 −15625.0 46656.0 −246868. 262144. 531441. −1.00000e6
\(n\): 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 30.14.a.f ✓ 1
3.b odd 2 1 90.14.a.c 1
5.b even 2 1 150.14.a.a 1
5.c odd 4 2 150.14.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.14.a.f ✓ 1 1.a even 1 1 trivial
90.14.a.c 1 3.b odd 2 1
150.14.a.a 1 5.b even 2 1
150.14.c.e 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 246868 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(30))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 64 \) Copy content Toggle raw display
$3$ \( T - 729 \) Copy content Toggle raw display
$5$ \( T + 15625 \) Copy content Toggle raw display
$7$ \( T + 246868 \) Copy content Toggle raw display
$11$ \( T + 5813808 \) Copy content Toggle raw display
$13$ \( T + 13914094 \) Copy content Toggle raw display
$17$ \( T + 81264378 \) Copy content Toggle raw display
$19$ \( T + 44056660 \) Copy content Toggle raw display
$23$ \( T - 68823216 \) Copy content Toggle raw display
$29$ \( T - 1046213610 \) Copy content Toggle raw display
$31$ \( T + 9196464568 \) Copy content Toggle raw display
$37$ \( T - 17229030002 \) Copy content Toggle raw display
$41$ \( T + 17123113638 \) Copy content Toggle raw display
$43$ \( T + 12353902084 \) Copy content Toggle raw display
$47$ \( T + 118626784848 \) Copy content Toggle raw display
$53$ \( T + 78925851894 \) Copy content Toggle raw display
$59$ \( T - 75016072560 \) Copy content Toggle raw display
$61$ \( T + 207562346458 \) Copy content Toggle raw display
$67$ \( T - 617859006332 \) Copy content Toggle raw display
$71$ \( T - 1849245670272 \) Copy content Toggle raw display
$73$ \( T - 1449378197666 \) Copy content Toggle raw display
$79$ \( T - 3278875539560 \) Copy content Toggle raw display
$83$ \( T - 576023687916 \) Copy content Toggle raw display
$89$ \( T - 1441392149490 \) Copy content Toggle raw display
$97$ \( T + 9161637335998 \) Copy content Toggle raw display
show more
show less